Waves

Fundamental Wave Characteristics

• Energy Transfer Mechanism: Waves transfer energy from one location to another through continuous oscillations without causing a permanent net displacement of the medium itself.

• Equilibrium Position: The resting, undisturbed center line or position of zero displacement from which wave oscillations occur.

• Crest: The highest point or maximum positive displacement of a wave above the equilibrium line.

• Trough: The lowest point or maximum negative displacement of a wave below the equilibrium line.

• Amplitude:

  • Verbatim Definition: The greatest distance from the equilibrium position (or the distance between the resting position and the crest or trough).

  • Standard Unit: Meters ($m$).

  • Energy Relationship: The greater the amplitude of a wave, the greater the amount of energy carried by that wave.

• Wavelength ($\lambda$):

  • Verbatim Definition: The shortest distance between points where the wave pattern repeats itself (or the distance from the top of one crest to the next consecutive crest, or from one trough to the next consecutive trough).

  • Symbol: The Greek letter lambda ($\lambda$).

  • Standard Unit: Meters ($m$).

• Frequency ($f$):

  • Verbatim Definition: The number of complete oscillations a point on that wave makes each second (or the number of wavelengths that pass by a given point in $1$ second).

  • Standard Unit: Hertz ($Hz$), equivalent to inverse seconds ($s^{-1}$ or $\frac{1}{s}$).

  • Proportional Comparisons:

  • $6$ waves passing a point in $1$ second corresponds to a frequency of $6\,Hz$ ($\frac{6}{1}$).

  • $3$ waves passing a point in $1$ second corresponds to a frequency of $3\,Hz$ ($\frac{3}{1}$).

• Period ($T$):

  • Verbatim Definition: The time it takes a wave to complete one full cycle or oscillation (the time required for a wave to go through one cycle).

  • Symbol: $T$.

  • Standard Unit: Seconds ($s$).

  • Mathematical Relationship with Frequency: Period and frequency are inversely proportional:

  • $T = \frac{1}{f}$

  • $f = \frac{1}{T}$

  • Calculated Examples from Frequency:

  • When frequency is $6\,Hz$, the period is $T = \frac{1}{6}\,s$.

  • When frequency is $3\,Hz$, the period is $T = \frac{1}{3}\,s$.

  • Inverse Scaling Behavior: If the frequency doubles ($f \rightarrow 2f$), the period is halved ($T \rightarrow \frac{T}{2}$). Conversely, if the period doubles, the frequency is halved.

Wavefronts and Rays

• Wavefront:

  • Verbatim Definition: An imaginary surface representing corresponding points of a wave that vibrate in unison.

  • Phase Alignment: When identical waves having a common origin travel through a homogeneous medium, corresponding crests and troughs at any given instant are completely in phase.

• Ray:

  • Definition and Function: A graphical simplification representing the trajectory of a wave by considering solely the direction that a particular section of the wave is traveling.

  • Path Formation: Joining these directional vectors traces out the linear path of a particular piece of a wave.

  • Geometric Relationship: Rays are always drawn perpendicular ($90^\circ$) to the wavefronts.

The Wave Equation Derivation and Relationships

• Kinematic Derivation:

  • Fundamental formula for constant speed:     $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$     $v = \frac{d}{t}$

  • Mapping kinematic quantities onto periodic wave parameters:

  • Speed corresponds to wave velocity $v$, measured in meters per second ($m\,s^{-1}$).

  • Distance corresponds to wavelength $\lambda$, measured in meters ($m$).

  • Time corresponds to period $T$, measured in seconds ($s$).

  • Substituting wave variables into the kinematic equation:     $v = \frac{\lambda}{T}$

  • Incorporating the inverse frequency relationship ($T = \frac{1}{f}$, where $f$ is measured in $s^{-1}$ or $Hz$):     $v = \frac{\lambda}{\frac{1}{f}} = f \times \lambda$

• Standard Wave Equation:   $v = f \times \lambda$

  • $v$ represents the speed or velocity of the wave ($m\,s^{-1}$).

  • $f$ represents the frequency of the wave ($Hz$ or $s^{-1}$).

  • $\lambda$ represents the wavelength of the wave ($m$).

• Algebraic Rearrangements:   $\lambda = \frac{v}{f}$   $f = \frac{v}{\lambda}$

Worked Mathematical Examples

• Example 1: Calculating Frequency and Period from Physical Oscillations

  • Problem Statement: A young girl on a swing completes $20.0$ cycles in $25$ seconds. Calculate the frequency and the period.

  • Frequency Solution:     $\text{Frequency} = \frac{\text{Number of cycles}}{\text{Time elapsed}}$     $f = \frac{20.0}{25} = 0.8\,Hz$

  • Period Solution:     $T = \frac{1}{f}$     $T = \frac{1}{0.8} = 1.25\,s$

• Example 2: Calculating Wave Speed from Frequency and Wavelength

  • Problem Statement: What is the speed of a wave that has a frequency of $5000\,Hz$ and a wavelength of $0.1\,m$?

  • Calculation:     $v = f \times \lambda$     $v = 5000 \times 0.1$     $v = 500\,m\,s^{-1}$

• Example 3: Calculating Wavelength from Speed and Frequency

  • Problem Statement: Calculate the wavelength (in meters) for a wave on a slinky spring with a frequency of $2\,Hz$ travelling at $3\,m\,s^{-1}$.

  • Calculation:     $v = f \times \lambda$     $\lambda = \frac{v}{f}$     $\lambda = \frac{3}{2} = 1.5\,m$

• Example 4: Calculating Velocity from Period and Wavelength

  • Problem Statement: What is the velocity of a wave that has a period of $0.01\,s$ and a wavelength of $2.0\,m$?

  • Frequency Determination:     $f = \frac{1}{T}$     $f = \frac{1}{0.01} = 100\,Hz$

  • Velocity Determination:     $v = f \times \lambda$     $v = 100 \times 2.0 = 200\,m\,s^{-1}$

Sound Waves, Wave Classification, and Applications

• Wave Classification by Motion:

  • Transverse Waves: Oscillations of the medium occur perpendicular ($90^\circ$) to the direction of wave propagation and energy transfer (e.g., surface ripples, slinky shear waves, electromagnetic waves).

  • Longitudinal Waves: Oscillations of the medium occur parallel to the direction of wave propagation, creating alternating regions of compression and rarefaction (e.g., sound waves, compression waves on a spring).

• Classification of Sound:

  • Loudness: Correlates directly with the amplitude of the sound wave. Larger amplitudes produce louder sounds carrying higher acoustic energy.

  • Pitch: Correlates directly with the frequency of the sound wave. Higher frequencies produce higher perceived pitch, whereas lower frequencies produce lower perceived pitch.

• Sound Medium Requirement and Vacuums:

  • Sound waves are mechanical longitudinal waves requiring physical particles to collide and transmit vibrational energy.

  • Sound cannot travel or be heard in a vacuum because there are no particles or physical medium present to facilitate mechanical energy transmission.

• Ultrasound:

  • High-frequency acoustic vibrations exceeding the upper threshold of human hearing ($20\,000\,Hz$ or $20\,kHz$).

• Echoes and Sonar Applications:

  • Echo Mechanics: An echo is formed when a sound wave reflects off a boundary or barrier and returns to the observer or receiver.

  • Sonar (Sound Navigation and Ranging): A technology utilizing pulses of sound sent underwater to measure water depth or detect submerged objects (such as submarines or sea beds).

  • Two-Way Path Calculation: In pulse-echo calculations, the wave travels to the reflective boundary and back, meaning total distance covered is twice the one-way distance ($2 \times d$):     $2 \times d = v \times t$     $d = \frac{v \times t}{2}$