NCEA Level 3 Physics: Wave Systems

Physics 3.3 Assessment Framework and Core Relationships

• Standard Classification and Metadata:

  • Subject Reference: Physics 3.3

  • Standard Title: Demonstrate understanding of wave systems

  • Level: Level 3

  • Credit Value: 4 Credits

  • Assessment Mode: External Assessment

• Achievement Criteria Levels:

  • Achievement: Demonstrate understanding of wave systems.

  • Achievement with Merit: Demonstrate in-depth understanding of wave systems.

  • Achievement with Excellence: Demonstrate comprehensive understanding of wave systems.

• Curriculum Content Scope and Wave System Phenomena:

  • Interference (Quantitative): Analysis of electromagnetic and sound waves, including two-point source interference, multi-slit interference patterns, and diffraction gratings.

  • Standing Waves: Development and characteristics of standing waves in strings and resonant air columns (pipes).

  • Acoustic Phenomena: Harmonics, overtones, resonance, and the generation of beats.

  • Doppler Effect: Examination of frequency shifts for mechanical waves involving a stationary observer and moving sources.

• Fundamental Quantitative Relationships:

  • Wave Velocity:     $v = f\lambda$

  • Wave Period and Frequency:     $f = \frac{1}{T}$

  • Angular Diffraction Grating Equation:     $d\sin(\theta) = n\lambda$

  • Double-Slit Small Angle Approximation:     $n\lambda = \frac{dx}{L}$

  • Doppler Frequency Shift (Moving Source, Stationary Observer):     $f' = f\left(\frac{v_w}{v_w \pm v_s}\right)$

  • Beat Frequency Relationship:     $|f_B| = |f_1 - f_2|$     (Note: While $|f_B| = |f_1 - f_2|$ is not explicitly printed in the standard formula sheet, it is necessary to explain and calculate beat phenomena).

Fundamental Properties of Waves

• Definition and Energy Propagation:

  • A wave is a regular, periodic vibration or oscillation that transfers energy from one location to another through a medium or vacuum without the bulk transport of matter.

  • The amount of energy transmitted by a wave is proportional to the square of its amplitude ($E \propto A^2$); waves with larger amplitudes carry significantly more energy.

• Mechanical Wave Classifications in Material Media:

  • Transverse Waves:

  • Particle displacement is perpendicular (at right angles) to the direction of wave propagation and energy transfer.

  • Can be observed using a stretched slinky spring on a smooth horizontal surface: keeping one end rigidly fixed and oscillating the free end from side to side produces a transverse wave.

  • Features alternating high points called crests and low points called troughs.

• Longitudinal Waves:

  • Particle displacement is parallel to the direction of wave propagation and energy transfer.

  • Can be observed using a slinky spring by keeping one end fixed and vigorously pushing and pulling the other end back and forth along the axis of the spring.

  • Features alternating zones of high particle concentration called compressions and low particle concentration called rarefactions.

• Kinematic and Wave Parameters:

  • Frequency ($f$):

  • Defined as the number of complete oscillations, cycles, or waves passing a fixed reference point per unit time.

  • Measured in hertz ($\text{Hz}$) or reciprocal seconds ($\text{s}^{-1}$).

  • Determined entirely by the frequency of the source generating the wave and remains invariant when transitioning between media.

  • Period ($T$):

  • Defined as the time required to complete one full cycle of oscillation.

  • Measured in seconds ($\text{s}$).

  • Inversely related to frequency:       $f = \frac{1}{T} \iff T = \frac{1}{f}$

  • Speed ($v$):

  • Defined as the distance travelled by the wavefront per unit time.

  • Measured in meters per second ($\text{m}\,\text{s}^{-1}$).

  • Dependent solely upon the physical properties of the transmission medium (such as elasticity, tension, density, and temperature).

  • Wavelength ($\lambda$):

  • Defined as the spatial distance between any two successive points in identical phase on a wave (e.g., crest to adjacent crest, trough to adjacent trough, or compression to adjacent compression).

  • Measured in meters ($\text{m}$).

  • Wave Equation:

  • Connects wave speed, frequency, and wavelength:       $v = f\lambda$

Electromagnetic and Sound Wave Characteristics

• Electromagnetic Waves:

  • Electromagnetic waves are transverse waves consisting of oscillating electric and magnetic fields perpendicular to each other and to the direction of wave travel.

  • They do not require a material medium and propagate through a vacuum at the speed of light ($c \approx 3.00 \times 10^8\,\text{m}\,\text{s}^{-1}$).

  • Speed varies when propagating through physical media depending on the medium's optical density and refractive index.

  • Electromagnetic Spectrum Hierarchy (Ordered from lowest frequency / longest wavelength to highest frequency / shortest wavelength):

  • Mnemonic aid: "Remember My Instructions Visible Under X-ray Glasses"

  • Radio waves (Lowest frequency, longest wavelength)

  • Microwaves

  • Infrared radiation

  • Visible Light (Spectrum ordered from long to short wavelength: Red, Orange, Yellow, Green, Blue, Indigo, Violet — ROYGBIV)

  • Ultraviolet radiation

  • X-rays

  • Gamma rays (Highest frequency, shortest wavelength)

• Sound Waves:

  • Sound waves are mechanical, longitudinal waves consisting of travelling compressions and rarefactions propagating through a material medium.

  • Cannot propagate through a vacuum.

  • Produced by mechanically vibrating objects:

  • Plucked strings (e.g., guitar), bowed strings (e.g., violin), or struck strings (e.g., piano).

  • Oscillating air columns within open or closed pipes, brass, and woodwind instruments.

  • Vibrating diaphragms, such as the back-and-forth oscillation of a loudspeaker cone.

• Acoustic Pressure Dynamics and Loudness:

  • A vibrating loudspeaker cone produces regions of elevated air pressure (compressions, $C$) and regions of reduced air pressure (rarefactions, $R$).

  • A spatial plot of instantaneous air pressure versus distance from the sound source forms a sinusoidal curve centered on baseline equilibrium atmospheric pressure.

  • Acoustic amplitude corresponds to the maximum displacement of particles from equilibrium, or equivalently, the maximum pressure difference above or below normal ambient atmospheric pressure ($\Delta P_{\max}$).

  • Greater wave amplitude corresponds directly to greater perceived loudness of the sound.

  • Speed of Sound in Various Media:

• Sound travels much slower than electromagnetic waves.

• Speed depends directly on the elasticity, compressibility, and density of the medium.

• In gases, sound speed increases with temperature because higher thermal kinetic energy enables faster transfer of momentum between colliding molecules.

Wave Interactions: Reflection, Refraction, and Diffraction

• Reflection:

  • Occurs when an incident wave strikes a boundary or barrier separating two media and bounces back into the original medium.

  • Invariance of Wave Parameters: Speed ($v$), frequency ($f$), period ($T$), and wavelength ($\lambda$) remain unchanged following reflection.

  • Modifications: The direction of travel is altered according to the law of reflection (angle of incidence equals angle of reflection). Depending on boundary rigidity, phase changes may occur (reflection from a denser or fixed medium induces an inversion / $180^\circ$ phase shift).

• Refraction:

  • Occurs when a wave crosses an interface from one material or medium into another medium with different physical characteristics.

  • Parameter Modifications: Wave speed ($v$) and wavelength ($\lambda$) change in direct proportion.

  • Invariance: Frequency ($f$) remains constant because it is determined by the oscillation rate of the original source:     $\frac{v_1}{\lambda_1} = \frac{v_2}{\lambda_2} = f$

  • Directional Deviation: The wave trajectory bends toward or away from the normal to the boundary depending on whether it decelerates or accelerates upon entering the second medium (unless the wave strikes normal to the surface, in which case direction does not alter).

• Diffraction:

  • Defined as the spreading or bending of wavefronts as they pass through an aperture (slit) or around the edges of an obstacle.

  • Condition for Maximum Diffraction: Diffraction effects are most pronounced when the aperture width or obstacle size is roughly equal to the wavelength of the wave:     $w \approx \lambda$

  • If an aperture is much wider than the wavelength ($w \gg \lambda$), the wave passes through with negligible spreading, casting a distinct geometric shadow.

  • Smaller obstacles and narrower slits produce greater angular bending for waves of a given wavelength.

  • Longer wavelengths bend significantly more around obstacles and spread wider through openings than shorter wavelengths.

• Everyday Acoustic and Optical Diffraction Comparison:

  • Audible sound waves possess wavelengths ranging from several centimeters to several meters ($\lambda \approx 0.02\,\text{m}\text{ to }17\,\text{m}$). Because these dimensions match everyday architectural features like doorways and windows ($w \approx 1\,\text{m}$), sound diffracts extensively, enabling humans to hear around corners.

  • Low-frequency sound (such as the deep bass notes produced by automobile audio systems) has long wavelengths and diffracts around vehicular obstacles and buildings far more efficiently than the short-wavelength, high-frequency melodies.

  • Visible light has extremely short wavelengths ($\lambda \approx 400\,\text{nm}\text{ to }700\,\text{nm} = 4 \times 10^{-7}\,\text{m}\text{ to }7 \times 10^{-7}\,\text{m}$). Because everyday apertures are orders of magnitude larger than $\lambda$, diffraction of light is negligible under ordinary conditions, preventing humans from seeing around corners.

• Single-Slit Diffraction Patterns:

  • Produced by self-interference between secondary Huygens wavelets originating across the single aperture.

  • Unlike pure two-source interference fringes, a single-slit diffraction pattern consists of a very broad, intense central maximum flanked by subsidiary fringes of progressively decreasing intensity and decreasing fringe width.

  • Monochromatic light comparison: Red light ($\lambda \approx 650\text{--}700\,\text{nm}$) has a longer wavelength than blue light ($\lambda \approx 450\text{--}480\,\text{nm}$). Consequently, red light generates a wider central maximum and more broadly spaced subsidiary bands than blue light.

Superposition and Wave Interference

• Principle of Superposition:

  • When two or more waves traverse the same spatial region simultaneously, the resultant displacement at any point and time is the algebraic sum of the individual displacements of the constituent waves.

  • Waves pass through one another without permanently altering their individual shapes, frequencies, or velocities.

• Types of Interference:

  • Constructive Interference:

  • Occurs when constituent waves arrive at a spatial point in phase (crests aligning with crests, or compressions aligning with compressions).

  • The combined displacement is greater than the displacement of the individual waves, producing a maximum resultant amplitude.

  • For sound, this results in increased volume; for light, it produces bright fringes.

• Destructive Interference:

  • Occurs when constituent waves arrive out of phase by half a cycle ($180^\circ$ or $\pi\,\text{radians}$), where a crest aligns with a trough, or a compression aligns with a rarefaction.

  • The opposing displacements cancel each other out algebraically, reducing or eliminating the resultant amplitude.

  • For sound, this produces silence or attenuation; for light, it produces dark fringes.

• Conditions for Sustained Interference Patterns (Coherence):

  • To create a stable, observable interference pattern over time, the wave sources must be coherent:

  • The emitting sources must have identical frequencies ($f_1 = f_2$) and identical wavelengths.

  • The emitting sources must maintain a constant, invariant phase relationship over time.

Quantitative Two-Slit Interference and Diffraction Gratings

• Young's Double-Slit Experiment:

  • Monochromatic light from a single source illuminates two closely spaced, narrow parallel slits ($S_1$ and $S_2$) separated by center-to-center distance $d$.

  • The two slits act as secondary coherent wave sources, emitting cylindrical wavefronts that overlap and interfere in the space beyond.

  • An observation screen positioned at a distance $L$ displays an alternating pattern of bright fringes (constructive interference maxima) and dark fringes (destructive interference minima).

• Mathematical Derivation and Path Difference Geometry:

  • Let $r_1$ and $r_2$ be the path lengths travelled by rays from slits $S_1$ and $S_2$ to an arbitrary point on the screen located distance $x$ from the central axis.

  • The geometric path difference ($\Delta r$) between the two rays when $d \ll L$ is:     $\Delta r = r_2 - r_1 = d\sin(\theta)$

  • Where $\theta$ is the angle between the emergent rays and the central normal axis.

  • When the fringe position $x$ is small compared to the distance to the screen $L$ ($x \ll L$), the small angle approximation applies:     $\sin(\theta) \approx \tan(\theta) = \frac{x}{L}$

  • Criteria for Constructive Interference (Bright Bands / Maxima):

  • Constructive interference requires the path difference to be an integer multiple of whole wavelengths:       $d\sin(\theta) = n\lambda \quad (n = 0, 1, 2, 3, \dots)$

  • Substituting the small angle approximation yields:       $n\lambda = \frac{dx}{L}$

    • Where $n$ represents the order number of the fringe ($n = 0$ corresponds to the central zero-order maximum, $n = 1$ to the first-order maximum, etc.).

  • Criteria for Destructive Interference (Dark Bands / Minima):

  • Destructive interference requires the path difference to be an odd half-integer multiple of wavelengths:       $d\sin(\theta) = \left(n + \frac{1}{2}\right)\lambda \quad (n = 0, 1, 2, 3, \dots)$

  • Using the small angle approximation:       $\left(n + \frac{1}{2}\right)\lambda = \frac{dx}{L}$

• Diffraction Gratings:

  • A diffraction grating consists of a large number of closely spaced, parallel, equidistant transmitting slits or reflecting grooves etched onto a flat substrate.

  • Slit spacing ($d$) is determined from the grating ruling density $N$ (number of lines per meter):     $d = \frac{1}{N}$

  • Each slit acts as an independent coherent source emitting wavefronts in phase with all other slits.

• Diffraction Grating Characteristics vs. Double-Slit Systems:

  • The large number of interfering slits causes destructive interference to occur almost everywhere between principal maxima.

  • The resulting interference pattern features exceptionally sharp, narrow, widely separated, and highly intense bright maxima set against dark backgrounds.

  • The angles to the principal maxima are determined by the grating equation:       $d\sin(\theta) = n\lambda$

  • (Note: Because angular deviation $\theta$ is typically large with diffraction gratings, the small-angle approximation $\sin(\theta) \approx \frac{x}{L}$ cannot be assumed; exact trigonometric relationships must be calculated).

  • Polychromatic (White Light) Dispersion via Diffraction Grating:

• Zero-Order Maximum ($n = 0$): At $\theta = 0^\circ$, the path difference for all wavelengths is zero ($\Delta r = 0$). All colors superimpose constructively in identical phase, producing a pure white central line.

• Higher-Order Maxima ($n \ge 1$): For any grating, slit spacing $d$ is fixed. From $\sin(\theta) = \frac{n\lambda}{d}$, the diffraction angle $\theta$ is proportional to wavelength $\lambda$:

  • Blue/violet light has shorter wavelengths ($\approx 400\text{--}450\,\text{nm}$) and is diffracted through smaller angles, appearing closest to the central axis.

  • Red light has longer wavelengths ($\approx 650\text{--}700\,\text{nm}$) and is diffracted through larger angles, appearing furthest from the central axis.

  • Each non-zero order produces a dispersed spectrum of colors from violet to red.

Resonance, Natural Frequencies, and Harmonic Series

• Resonance Dynamics:

  • Every elastic mechanical system possesses one or more natural frequencies of vibration determined by its physical dimensions, mass distribution, and restoring forces.

  • When an external periodic driving force or periodic impulse is applied to an object at a frequency matching one of its natural resonant frequencies, energy transfer is maximized.

  • Continued application of impulses in phase with the natural frequency causes the vibration amplitude to grow.

• Practical Manifestations of Mechanical and Acoustic Resonance:

  • Child on a Swing: Repeated pushes timed to match the natural pendulum frequency of the swing transfer energy constructively, increasing the oscillation amplitude.

  • Structural Failure of Bridges: Bridges exhibit natural structural periods of vibration. Marching soldiers break stride when crossing bridges because synchronized footsteps matching the natural frequency could drive large-amplitude oscillations, risking structural collapse.

  • Sympathetic Vibration in Strings: Singing a sustained pitch near an acoustic piano excites resonant vibrations in the specific piano string tuned to that exact frequency.

  • Acoustic Glass Shattering: A singer holding a loud, sustained note matching the natural vibrational resonance of a crystal wine glass can induce vibrations that exceed the tensile strength of the glass, causing it to shatter.

• Harmonics and Overtones:

  • Fundamental Frequency ($f_0$ or $f_1$): The lowest frequency of standing wave that can be sustained within a vibrating system, corresponding to the longest possible wavelength.

  • Harmonics: Frequencies that are exact integer multiples of the fundamental frequency ($f_n = n f_1$, where $n = 1, 2, 3, \dots$).

  • Overtones: Any resonant frequency higher than the fundamental. The first overtone is the next permissible resonant mode above the fundamental; the second overtone is the mode above that.

  • Harmonic Series Nomenclature:

  • $f_1$ = Fundamental frequency = First harmonic

  • $2f_1$ = Second harmonic = First overtone

  • $3f_1$ = Third harmonic = Second overtone

  • $n f_1$ = $n$-th harmonic = $(n-1)$-th overtone

  • Musical Timbre (Sound Quality): The characteristic tone of an instrument is determined by the relative amplitudes of the various harmonics in its acoustic spectrum. Adjusting string tension shifts pitch (greater tension increases wave speed $v = \sqrt{\frac{T}{\mu}}$, elevating frequency for a fixed length), while blowing harder on a pipe excites higher-order harmonic modes.

Standing Waves on Stretched Strings

• Formation Mechanism:

  • A standing (stationary) wave is formed by the continuous superposition of two travelling waves of identical frequency, wavelength, and amplitude propagating in opposite directions through the same medium.

  • On a string fixed at both ends, a wave pulse travels down the string, reflects off the boundary, and inverts ($180^\circ$ phase shift). The reflected wave superimposes with oncoming incident waves.

• Spatial Anatomy of Standing Waves:

  • Nodes ($N$): Positions along the medium where incident and reflected waves always interfere completely destructively (equal magnitude, opposite phase). Net displacement is permanently zero.

  • Antinodes ($A$ or $AN$): Positions along the medium where incident and reflected waves interfere constructively, producing oscillations of maximum amplitude.

  • Spatial Relationships:

  • Distance between two adjacent nodes ($N \leftrightarrow N$): $\frac{\lambda}{2}$

  • Distance between two adjacent antinodes ($A \leftrightarrow A$): $\frac{\lambda}{2}$

  • Distance between an adjacent node and antinode ($N \leftrightarrow A$): $\frac{\lambda}{4}$

• Resonant Modes for Strings of Length $L$ Fixed at Both Ends:

  • Boundary condition: Fixed terminations cannot move; therefore, a node must exist at each fixed end.

• First Harmonic (Fundamental, $n = 1$):

  • Node at each end with a single central antinode ($N-A-N$).

  • Length-wavelength relationship:     $L = \frac{\lambda_1}{2} \implies \lambda_1 = 2L$

  • Resonant frequency:     $f_1 = \frac{v}{\lambda_1} = \frac{v}{2L}$

• Second Harmonic (First Overtone, $n = 2$):

  • Three nodes and two antinodes ($N-A-N-A-N$).

  • Length-wavelength relationship:     $L = \lambda_2 \implies \lambda_2 = L = \frac{2L}{2}$

  • Resonant frequency:     $f_2 = \frac{v}{\lambda_2} = \frac{v}{L} = 2f_1$

• Third Harmonic (Second Overtone, $n = 3$):

  • Four nodes and three antinodes ($N-A-N-A-N-A-N$).

  • Length-wavelength relationship:     $L = \frac{3\lambda_3}{2} \implies \lambda_3 = \frac{2L}{3}$

  • Resonant frequency:     $f_3 = \frac{v}{\lambda_3} = \frac{3v}{2L} = 3f_1$

• Fourth Harmonic (Third Overtone, $n = 4$):

  • Five nodes and four antinodes.

  • Length-wavelength relationship:     $L = \frac{4\lambda_4}{2} = 2\lambda_4 \implies \lambda_4 = \frac{2L}{4} = \frac{L}{2}$

  • Resonant frequency:     $f_4 = 4f_1$

• General Equations for Strings:   $\lambda_n = \frac{2L}{n} \quad \text{and} \quad f_n = n f_1 = n\left(\frac{v}{2L}\right) \quad (n = 1, 2, 3, 4, \dots)$

Standing Waves in Resonant Air Columns (Pipes)

• Acoustic Boundary Conditions:

  • Closed End of a Tube: Air molecules are blocked by a rigid barrier and cannot vibrate longitudinally. A displacement node ($N$) must form at a closed end.

  • Open End of a Tube: Air molecules are exposed to the atmosphere and can vibrate with maximum longitudinal amplitude. A displacement antinode ($A$) must form at an open end.

• Pipes Open at Both Ends (Open Pipes):

  • Boundary condition: An antinode must form at each open end ($A \leftrightarrow A$).

  • First Harmonic (Fundamental, $n = 1$):

  • Contains an antinode at each end and one central node ($A-N-A$).

  • Length-wavelength relation:       $L = \frac{\lambda_1}{2} \implies \lambda_1 = 2L$

  • Frequency:       $f_1 = \frac{v}{2L} = f$

  • Second Harmonic (First Overtone, $n = 2$):

  • Contains two nodes and three antinodes ($A-N-A-N-A$).

  • Length-wavelength relation:       $L = \lambda_2 \implies \lambda_2 = L = \frac{2L}{2}$

  • Frequency:       $f_2 = 2f_1 = 2f$

  • Third Harmonic (Second Overtone, $n = 3$):

  • Contains three nodes and four antinodes ($A-N-A-N-A-N-A$).

  • Length-wavelength relation:       $L = \frac{3\lambda_3}{2} \implies \lambda_3 = \frac{2L}{3}$

  • Frequency:       $f_3 = 3f_1 = 3f$

  • Fourth Harmonic (Third Overtone, $n = 4$):

  • Length-wavelength relation:       $L = \frac{4\lambda_4}{2} \implies \lambda_4 = \frac{2L}{4} = \frac{L}{2}$

  • Frequency:       $f_4 = 4f_1 = 4f$

  • Open Pipe Summary: Open pipes support all integer harmonics ($n = 1, 2, 3, 4, \dots$):     $\lambda_n = \frac{2L}{n} \quad \text{and} \quad f_n = n f_1$

• Pipes Closed at One End (Closed Pipes):

  • Boundary condition: A displacement node must form at the closed bottom, and an antinode must form at the open top ($N \leftrightarrow A$).

  • First Harmonic (Fundamental, $n = 1$):

  • Contains one node at the closed end and one antinode at the open end ($N-A$).

  • Length-wavelength relation:       $L = \frac{\lambda_1}{4} \implies \lambda_1 = 4L$

  • Frequency:       $f_1 = \frac{v}{4L} = f$

  • Third Harmonic (First Overtone, $n = 3$):

  • Contains two nodes and two antinodes ($N-A-N-A$).

  • Length-wavelength relation:       $L = \frac{3\lambda_3}{4} \implies \lambda_3 = \frac{4L}{3}$

  • Frequency:       $f_3 = 3f_1 = 3f$

  • Fifth Harmonic (Second Overtone, $n = 5$):

  • Contains three nodes and three antinodes ($N-A-N-A-N-A$).

  • Length-wavelength relation:       $L = \frac{5\lambda_5}{4} \implies \lambda_5 = \frac{4L}{5}$

  • Frequency:       $f_5 = 5f_1 = 5f$

  • Even Harmonic Absence: Closed pipes cannot produce even harmonics ($2nd, 4th, 6th, \dots$). Because one end is fixed as a node and the other as an antinode, the length of the column must contain an odd number of quarter wavelengths:     $L = \frac{m\lambda}{4} \quad (m = 1, 3, 5, 7, \dots)$

  • Closed Pipe General Expressions:     $\lambda_m = \frac{4L}{m} \quad \text{and} \quad f_m = m f_1 = m\left(\frac{v}{4L}\right) \quad (m = 1, 3, 5, 7, \dots)$

• Acoustic End Correction:

  • The antinode at the open end of a pipe does not form precisely at the physical rim; it forms a small distance beyond the tube opening in the outside air.

  • In precise calculations, an effective acoustic length ($L_{\text{eff}} = L_{\text{physical}} + c_{\text{correction}}$) is used to account for this extra distance.

Acoustics of Beats

• Physical Principle of Beats:

  • Beats occur when two waves of the same type and nearly equal frequencies ($f_1 \approx f_2$) travel through the same region and superimpose.

  • Because the frequencies differ slightly, the relative phase between the two waves changes periodically:

  • When the two waves cycle into phase, they interfere constructively, producing a maximum amplitude and louder sound.

  • When the two waves cycle $180^\circ$ out of phase, they interfere destructively, producing a minimum amplitude and softer sound or silence.

  • This periodic rise and fall in volume is heard as a regular pulsing sound.

• Quantitative Beat Formulation:

  • Beat Frequency ($f_B$): The number of volume pulses heard per second equals the absolute difference between the constituent frequencies:     $|f_B| = |f_1 - f_2|$

  • Carrier / Average Frequency ($f_{\text{avg}}$): The pitch perceived by the human ear corresponds to the mean frequency of the two sources:     $f_{\text{average}} = \frac{f_1 + f_2}{2}$

  • Numerical Example: If a sound wave of $440\,\text{Hz}$ superimposes with a sound wave of $444\,\text{Hz}$, the beat frequency is:     $|f_B| = |440 - 444| = 4\,\text{Hz}$

  • The observer hears a $442\,\text{Hz}$ tone pulsating in loudness 4 times per second.

• Auditory Perception and Tuning:

  • Musical Tuning: Musicians use beats to tune instruments like guitars. Striking an out-of-tune string alongside a reference pitch produces beats. As the string is tightened or loosened, the frequency difference shrinks, slowing the beat rate until it reaches zero ($f_B = 0\,\text{Hz}$), indicating the string is in tune.

  • Auditory Limits and Discord: The human ear perceives beats as distinct volume variations only when the frequencies are very close. When the frequency difference is large, the beats occur too rapidly to be heard as distinct pulses.

  • For example, sounding the notes C ($262\,\text{Hz}$) and B ($247\,\text{Hz}$) simultaneously yields a beat frequency of:     $f_B = 262 - 247 = 15\,\text{Hz}$

  • A modulation rate of 15 beats per second is perceived not as distinct beats, but as an unpleasant acoustic discord (dissonance).

The Doppler Effect

• Physical Concept:

  • The Doppler effect is the apparent change in the observed frequency of a wave caused by relative motion between the wave source and an observer.

  • When the source moves toward an observer, the observed frequency is higher than the emitted frequency ($f' > f$).

  • When the source moves away from an observer, the observed frequency is lower than the emitted frequency ($f' < f$).

  • An everyday example is the pitch shift of a moving car horn: the horn sounds higher-pitched as the car approaches and drops to a lower pitch as the car speeds away.

• Mechanism for a Moving Source and Stationary Observer:

  • Wave propagation speed ($v_w$) through the medium is independent of the motion of the source and remains constant.

  • Ahead of the Moving Source: The source pursues the wavefronts it emits, compressing the waves into a smaller space. This shortens the apparent wavelength ($\lambda' < \lambda$), so wavefronts arrive at a stationary observer more frequently, increasing the perceived pitch ($f' > f$).

  • Behind the Moving Source: The source moves away from the wavefronts it emits, stretching them out over a larger space. This lengthens the apparent wavelength ($\lambda' > \lambda$), so wavefronts arrive at a stationary observer less frequently, decreasing the perceived pitch ($f' < f$).

  • Higher source speeds ($v_s$) produce greater Doppler shifts in wavelength and frequency.

• Derivation of the Doppler Equation (Stationary Observer):

  • Let the stationary source emit waves with frequency $f_s$, period $T_s = \frac{1}{f_s}$, and velocity $v_w$.

  • In one period $T_s$, a wavefront travels a distance of $\lambda = v_w T_s$.

  • During that same period $T_s$, the source moves toward or away from the observer by a distance of $\Delta d = v_s T_s$.

  • The observed wavelength ($\lambda'$) becomes:     $\lambda' = \lambda \mp v_s T_s = v_w T_s \mp v_s T_s = (v_w \mp v_s)T_s = \frac{v_w \mp v_s}{f_s}$

  • From the general wave equation, the wave speed relative to the stationary observer remains $v_w$:     $v_w = f'\lambda' \implies \lambda' = \frac{v_w}{f'}$

  • Equating the two expressions for observed wavelength ($\lambda'$):     $\frac{v_w}{f'} = \frac{v_w \pm v_s}{f_s}$

  • Cross-multiplying:     $f'(v_w \pm v_s) = f_s v_w$

  • Solving for the observed frequency ($f'$):     $f' = f_s\left(\frac{v_w}{v_w \pm v_s}\right)$

  • Sign Conventions:

  • Use $(v_w - v_s)$ in the denominator when the source is moving towards the observer (the smaller denominator yields an increased frequency, $f' > f_s$).

  • Use $(v_w + v_s)$ in the denominator when the source is moving away from the observer (the larger denominator yields a decreased frequency, $f' < f_s$).

• Applications and Assessment Contexts:

  • Police Radar Speed Traps: A radar device emits a signal at a known frequency toward an oncoming vehicle. The reflected signal returns with a Doppler-shifted frequency. The radar unit superimposes the outgoing and reflected waves, extracts the resulting beat frequency ($f_B$), and computes the vehicle's speed.

• Astronomical Spectroscopy: The Doppler effect shifts the absorption and emission spectra of stars and distant galaxies. Spectral lines shifted toward longer, redder wavelengths (redshift) indicate motion away from Earth, providing evidence for the expansion of the universe. Shifts toward shorter, bluer wavelengths (blueshift) indicate motion toward Earth.

• NCEA Curriculum Scope Note: The standard focuses primarily on mechanical waves (sound) involving stationary observers. Electromagnetic Doppler applications (such as cosmic redshift and radar speed detection) serve as real-world illustrations.

• Cultural Reference: A well-known popular culture depiction appears in The Big Bang Theory, where Sheldon Cooper wears a Doppler effect costume consisting of alternating white and dark concentric stripes while imitating the sound of a passing source ("Neeeeoooowwwww!").