HL IB Physics: Wave Phenomena Comprehensive Notes

Wavefronts and Rays

Waves are capable of propagating in both two and three dimensions. A surface wave, such as one on the surface of water, propagates in two dimensions and possesses circular wavefronts. In contrast, a spherical wave, such as sound or light, propagates in three dimensions and possesses spherical wavefronts. A circle is a 2D shape, while a sphere is a 3D shape, which is a critical distinction to remember for examination purposes.

Waves can be represented graphically using two primary methods. Wavefronts are defined as lines that join all the points of a wave that oscillate in phase; these are always perpendicular to the direction of motion and the direction of energy transfer. Rays are lines that explicitly show the direction of motion and energy transfer of the wave, and they are drawn perpendicular to the wavefront. For transverse waves traveling in a horizontal plane, wavefronts are typically viewed from above as a series of parallel vertical lines. In these diagrams, peaks are often represented by darker lines, while troughs are represented by fainter lines. Some diagrams may use only peak wavefronts. The distance between two successive peak wavefronts, or two successive trough wavefronts, is equal to the wavelength ($\lambda$) of the wave. When sketching or interpreting wavefronts and rays in an exam, it is vital to use a ruler to ensure lines are straight, as unclear or sloping diagrams are unlikely to receive full marks.

Reflection, Refraction, and Transmission

When waves reach a boundary between two different materials, they may undergo reflection, refraction, transmission, or absorption. In the context of optics, a transparent material is referred to as a medium, and the plural form is media. Reflection occurs when a wave hits a boundary between two media and, instead of passing through, bounces back into the original medium. The law of reflection states that the angle of incidence ($i$) is equal to the angle of reflection ($r$), or $i = r$. During reflection, the frequency, wavelength, and speed of the wave remain unchanged, though some of the wave energy may be absorbed or transmitted through the medium.

Refraction is defined as the change in direction of a wave when it passes through a boundary between media of different densities. This change in direction is caused by a change in the speed of different parts of the wavefront as they intersect the boundary. When a wave travels from a less dense medium into a denser medium (e.g., air into glass), the speed of the wave decreases, the angle of refraction becomes smaller, and the light bends towards the normal line, which is drawn at $90^{\circ}$ to the boundary. Conversely, when traveling from a denser medium to a less dense medium (e.g., glass into air), the wave speed increases, the angle of refraction becomes larger, and the light bends away from the normal. When a wave refracts, its speed ($v$) and wavelength ($\lambda$) change, but its frequency ($f$) remains constant, which is evidenced by the fact that the color of the wave does not change. If a ray is incident on a boundary at exactly $90^{\circ}$ (along the normal), the wave passes straight through without a change in direction because the entire wavefront enters the boundary at the same time and speed.

Refraction also occurs in water waves due to changes in depth. When waves pass from deep to shallow water, increased friction with the sea bed and less space for oscillation cause the waves to slow down. This results in a decrease in wavelength ($\lambda_{W} < \lambda_{A}$) and a reduced distance between peaks. If these waves hit the boundary at an angle, they refract towards the normal, meaning the angle of refraction is less than the angle of incidence ($r < i$). Transmission is a more general term for a wave appearing on the opposite side of a boundary. Refraction is a specific type of transmission involving a change in direction. During general transmission, the frequency or speed of the wave does not change, although the amplitude of the transmitted wave will be lower if partial absorption has occurred. These behaviors apply to both transverse and longitudinal waves. For conceptual clarity, imagine a car driving at an angle from a road onto mud; the wheel hitting the mud first slows down while the other maintains speed, causing the car to turn. In complex scenarios with two boundaries, the refracted ray from the first boundary becomes the incident ray at the second.

Mathematical Principles of Refraction

Snell\u2019s Law relates the angle of incidence to the angle of refraction. The absolute refractive index ($n$) of a material is a dimensionless ratio that indicates its optical density and is calculated as $n = \frac{c}{v}$, where $c$ is the speed of light in a vacuum ($3.00 \times 10^8\,m/s$) and $v$ is the speed of light in the medium. The refractive index of air is taken as $n = 1$. Higher refractive indices indicate more optically dense materials where light travels more slowly. Snell\u2019s Law is expressed as $\frac{n_1}{n_2} = \frac{\sin(\theta_2)}{\sin(\theta_1)} = \frac{v_2}{v_1}$, or in the more convenient form: $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$.

As the angle of incidence increases, the angle of refraction also increases until it reaches $90^{\circ}$. At this point, the light refracts along the boundary, and the angle of incidence is called the critical angle ($\theta_c$). The critical angle is found using the equation $\sin(\theta_c) = \frac{n_2}{n_1}$. Total Internal Reflection (TIR) occurs only when two conditions are met: the refractive index of the second medium must be less than the first ($n_2 < n_1$), and the angle of incidence must exceed the critical angle ($\theta_i > \theta_c$). TIR follows the law of reflection where the angle of incidence equals the angle of reflection. For example, light traveling from a material with $n = 1.2$ into air ($n = 1.0$) has a critical angle calculated from $\sin(\theta_c) = 0.83$, giving $\theta_c = 56^{\circ}$.

Diffraction of Waves

Diffraction is the spreading out of waves as they pass through a narrow gap (aperture) or around an obstruction (barrier). When waves pass through a gap, they spread out and develop curvature; however, the amplitude decreases because the barrier edges absorb some wave energy. The extent of diffraction is determined by the relationship between the wavelength ($\lambda$) and the gap width. Maximum diffraction occurs when the wavelength is comparable to or larger than the gap width. If the gap is much smaller than the wavelength, the wave passes over it without diffracting. If the gap is much wider than the wavelength, the majority of the wave passes straight through with minimal diffraction. For instance, an electric guitar student practicing in a room with a $10\,cm$ ($0.1\,m$) door gap will find that sound frequencies ($f$) best diffracted are those where $f \leq \frac{v}{\lambda}$, which for a sound speed of $340\,m/s$ equals $f \leq 3400\,Hz$.

Diffraction also occurs around barriers. If the barrier is larger than the wavelength, some diffraction occurs, but a significant shadow region exists behind the object, and many waves are reflected. If the barrier is the same size as the wavelength, more diffraction occurs, and the shadow region shrinks. If the barrier is smaller than the wavelength, the transcript states no diffraction occurs and the shadow region is very small. In a comparative scenario, X-rays ($1 \times 10^{-8}$ to $4 \times 10^{-13}\,m$) are best demonstrated to diffract through crystalline solids because the gap between atoms is comparable to their wavelength. Conversely, UV radiation is not diffracted by gate posts, and radio waves are not diffracted by human hair because their scales are mismatched.

Superposition and Interference

Superposition occurs when two or more waves arrive at the same point and overlap, causing their amplitudes to combine. The Principle of Superposition states that the total displacement at any point is the algebraic sum of the displacements of the individual waves. This is visible in surface water waves, where overlapping waves created by dogs swimming can result in areas of zero displacement (flat water) and areas of increased displacement (higher peaks). Individual displacements can be positive or negative, combining as vector quantities. After interacting, pulses continue their path undisturbed.

Interference is the observable effect of superposition. Constructive interference occurs when waves meet in phase (peak-to-peak or trough-to-trough), resulting in a larger resultant amplitude. Destructive interference occurs when waves meet in antiphase (peak-to-trough), causing them to cancel each other out. For interference to be stable and observable, waves must be coherent, meaning they possess the same frequency and a constant phase difference. Examples of coherent sources include monochromatic laser light or two speakers emitting the same frequency.

Whether interference is constructive or destructive depends on the path difference, which is the difference in distance traveled by two waves from their sources ($S_1$ and $S_2$) to a meeting point ($P$). Constructive interference occurs when the path difference is a whole number of wavelengths: $\text{path difference} = n\lambda$, where $n = 0, 1, 2, 3...$. Destructive interference occurs when the path difference is an odd number of half-wavelengths: $\text{path difference} = (n + \frac{1}{2})\lambda$. On wavefront diagrams, constructive interference is found where two peaks or two troughs intersect, while destructive interference occurs where a peak meets a trough.

Young\u2019s Double-Slit Experiment

Young\u2019s double-slit experiment demonstrates interference using a single monochromatic, coherent light source (typically a laser) passing through two narrow slits. The light diffracts at the slits and overlaps on a faraway screen. Constructive interference forms bright strips called fringes or maxima, while destructive interference forms dark strips called minima. All bright fringes in a standard double-slit pattern are identical in width and intensity. Each fringe represents a specific order ($n$); $n=0$ is the central maximum, while $n=1$ represents the first maximum on either side.

The separation between successive fringes ($s$) can be calculated using the double-slit equation: $s = \frac{\lambda D}{d}$, where $\lambda$ is the wavelength, $D$ is the distance from the slits to the screen, and $d$ is the separation between the two slits. Fringe separation increases if the wavelength increases, if the distance to the screen increases, or if the slit separation decreases. For a laser emitting at $750\,THz$, the wavelength is first calculated using $\lambda = \frac{c}{f}$. If the separation between maxima $P$ and $Q$ is $15\,mm$ and the screen distance is $4.5\,m$, these values allow for the calculation of the slit separation $d$. It is crucial not to confuse the slit separation ($d$) with the fringe separation ($s$) or the screen distance ($D$).

Single-Slit Diffraction (HL)

When monochromatic light passes through a single rectangular slit, it produces a diffraction pattern consisting of a wide, bright central maximum flanked by narrower, dimmer secondary maxima. The dark fringes are regions of zero intensity caused by destructive interference. The central maximum in a single-slit pattern is much wider than those in a double-slit pattern. Red light, having a longer wavelength, produces wider fringes because the angle of diffraction is greater. Conversely, blue light produces narrower fringes. If the slit width ($b$) is reduced, the angle of diffraction increases, the intensity of maxima decreases, and the width of the central maximum increases.

The angle of diffraction ($\theta$) for the first minimum is given by $\theta = \frac{\lambda}{b}$. This is derived by considering two parallel paths traveling at an angle $\theta$ with a path difference of $\frac{b}{2} \sin(\theta)$. Setting this to $\frac{\lambda}{2}$ for destructive interference and using the small-angle approximation ($\sin(\theta) \approx \theta$) results in the formula. Consequently, the width of the central maximum can be increased by reducing the slit width ($b$), increasing the wavelength ($\lambda$), or moving the screen further away ($D$). In practical scenarios, double-slit interference patterns are actually "modulated" by the single-slit intensity pattern. This means the equally spaced double-slit fringes appear inside an "envelope" defined by the single-slit pattern, assuming the slit width is not negligible.

Diffraction Gratings (HL)

A diffraction grating consists of a large number of very thin, equally spaced parallel slits. It produces sharper, narrower, and brighter fringes than a double-slit setup because more slits allow for more diffraction and constructive interference. The location of maxima is determined by the diffraction grating equation: $d \sin(\theta) = n\lambda$, where $d$ is the slit spacing and $n$ is the order. The slit spacing $d$ is related to the number of lines per meter ($N$) by the formula $d = \frac{1}{N}$. Angular separation ($\theta$) grows as the order $n$ increases. The highest order visible is limited by the fact that $\sin(\theta)$ cannot exceed $1$, so $n_{max} = \frac{d}{\lambda}$. Any calculated decimal value for $n$ must be rounded down (e.g., $n = 2.7$ means $n = 2$ is the highest order).

When white light is passed through a grating, the central maximum ($n=0$) appears as a thin white strip because all wavelengths interfere constructively there. All other maxima ($n=1, 2...$) are composed of a spectrum of colors. In these spectra, violet light is diffracted the least and appears closest to the center, while red light is diffracted the most and appears furthest away. As the orders increase, the spectra eventually merge as the fringe spacing decreases. In a worked example with a slit spacing of $1.7\,\mu m$ and a wavelength of $550\,nm$, the angle $\alpha$ between the two second-order lines is found by calculating $2\theta$ for $n=2$, ensuring care is taken not to confuse the angle from the center ($\theta$) with the total angular separation between orders.