IB Physics: Waves - Properties and Behaviour

Waves are the fundamental mechanism for transferring energy and information without transferring matter, making them one of the most pervasive concepts in physics. For the IB Physics student, a deep understanding of wave properties and behaviours is essential, as it connects directly to topics from sound and light to quantum mechanics.

1. Fundamental Wave Types and Properties

A wave is defined as an oscillation that travels through space and time, transferring energy from one point to another. We classify waves primarily by the direction of their oscillation relative to the direction of energy transfer.

Transverse waves are those where the particles of the medium oscillate perpendicular to the direction of the wave's energy transfer. The classic example is a wave on a string, where the string moves up and down while the wave travels horizontally. Electromagnetic waves, including light, are also transverse waves. In contrast, longitudinal waves feature oscillations parallel to the direction of energy transfer. Sound waves in air are longitudinal, where air particles are compressed and rarefied along the same line the sound is moving.

To describe these waves, we use a set of key properties. The wavelength ($\lambda$) is the distance between two consecutive identical points on a wave, such as crest-to-crest. The frequency ($f$) is the number of complete wave cycles passing a point per second, measured in Hertz (Hz). The wave speed ($v$) is the speed at which the wavefront, an imaginary surface connecting all points of the same phase, travels through the medium. These three quantities are intimately connected by the universal wave equation: $$v=f\lambda$$ This equation is a cornerstone of wave physics. For example, if a sound wave has a frequency of 500 Hz and a speed of $340{\text{ m s}}^{-1}$, its wavelength is calculated as $\lambda =v/f=340/500=0.68\text{ m}$.

2. Wave Behaviour at Boundaries: Reflection and Refraction

When a wave encounters a boundary between two different media, several things can happen. Reflection occurs when a wave bounces off a barrier. The law of reflection states that the angle of incidence equals the angle of reflection, with both angles measured from the normal (a line perpendicular to the surface). This is true for both light and sound.

Refraction is the change in direction and speed of a wave as it passes from one medium into another where its speed is different. This bending occurs because while the frequency of the wave remains constant, its speed and wavelength change. The relationship governing refraction is Snell's Law: $$n_1\sin {\theta }_1=n_2\sin {\theta }_2$$ Here, $n$ is the refractive index of a medium, defined as $n=c/v$, where $c$ is the speed of light in a vacuum and $v$ is the speed of light in the medium. ${\theta }_1$ and ${\theta }_2$ are the angles of incidence and refraction, respectively.

Step-by-step example: Light travels from air ($n_1=1.00$) into water ($n_2=1.33$) at an angle of incidence of $30^{\circ }$. Find the angle of refraction.

1. Apply Snell's Law: $1.00\times \sin (30^{\circ })=1.33\times \sin ({\theta }_2)$.
2. $\sin (30^{\circ })=0.5$, so $0.5=1.33\sin ({\theta }_2)$.
3. $\sin ({\theta }_2)=0.5/1.33\approx 0.376$.
4. ${\theta }_2={\sin }^{-1}(0.376)\approx 22^{\circ }$.

The light bends towards the normal because it is entering a medium with a higher refractive index where it travels slower.

3. Diffraction and the Principle of Superposition

Diffraction is the spreading out of waves as they pass through an aperture or around an obstacle. The amount of diffraction is significant when the width of the gap is comparable to the wavelength of the wave. This is why you can hear sound (long wavelength) around corners but cannot see light (very short wavelength) around the same corner. In IB, you will often analyze diffraction patterns from single slits and diffraction gratings.

When two or more waves meet, they combine. The principle of superposition states that the resultant displacement at any point is the vector sum of the displacements of the individual waves at that point. This leads directly to interference.

Constructive interference occurs when waves meet in phase (crest aligns with crest). The path difference between the waves is an integer multiple of the wavelength: $\text{path difference}=n\lambda$, where $n=0,1,2,...$. This results in a wave of increased amplitude. Destructive interference occurs when waves meet completely out of phase (crest aligns with trough). The condition is $\text{path difference}=(n+\frac{1}{2})\lambda$, resulting in a reduced or zero amplitude. These principles explain the interference patterns observed in double-slit experiments, a key topic in IB Physics.

4. Standing Waves and Resonance

A standing wave (or stationary wave) is formed when two identical waves travelling in opposite directions interfere continuously. Characteristic points called nodes (points of zero amplitude) and antinodes (points of maximum amplitude) appear at fixed positions. Standing waves are fundamental to understanding musical instruments.

They form under specific conditions related to resonance, which is the large-amplitude oscillation of a system when it is driven at its natural frequency. For a string fixed at both ends or an air column in a pipe, standing waves only form at specific resonant frequencies. The simplest pattern is the fundamental frequency (first harmonic). The conditions are:

• String fixed at both ends: $L=n(\frac{{\lambda }_n}{2})$, where $n=1,2,3,...$ and $L$ is the string length.
• Pipe open at both ends: same condition as the string.
• Pipe closed at one end: $L=n(\frac{{\lambda }_n}{4})$, where $n=1,3,5,...$ (only odd harmonics).

For example, the fundamental frequency ($n=1$) for a string of length $L$ is given by $f_1=v/(2L)$, where $v$ is the wave speed on the string. The ability to calculate harmonic frequencies is a frequent exam requirement.

Common Pitfalls

1. Confusing Wave Types with Wave Categories: A common error is stating that "all electromagnetic waves are longitudinal" or that "sound is transverse." Remember: transverse vs. longitudinal describes the oscillation direction. Electromagnetic waves are transverse; sound waves in fluids are longitudinal. Some waves, like seismic S-waves and P-waves, exemplify both types.

1. Misapplying the Wave Equation and Snell's Law: Students often forget that in $v=f\lambda$, the wave speed $v$ is determined by the medium. When a wave changes medium, $f$ stays constant, but $v$ and $\lambda$ change. In Snell's Law, a similar error is using angles measured from the surface instead of from the normal. Always sketch the normal line on diagrams.

1. Mixing Up Conditions for Interference: The conditions for constructive ($n\lambda$) and destructive ($(n+\frac{1}{2})\lambda$) interference are often reversed. A mnemonic: "Constructive is a full number of wavelengths, destructive is a half." Also, ensure the path difference is measured from the coherent sources to the point of interest.

1. Incorrect Harmonics for Closed Pipes: For a pipe closed at one end, only odd harmonics are present ($n=1,3,\mathrm{5...}$). A frequent mistake is to include the even harmonics ($n=2,\mathrm{4...}$). Remember the boundary condition: a closed end must be a node, an open end an antinode. This only allows for odd multiples of the quarter-wavelength to fit.

Summary

• Core Classification: Waves are classified as transverse (oscillation perpendicular to energy direction, e.g., light) or longitudinal (oscillation parallel, e.g., sound), and are described by wavelength ($\lambda$), frequency ($f$), and speed ($v$), related by $v=f\lambda$.
• Boundary Behaviours: Waves reflect (angle of incidence = angle of reflection) and refract. Refraction is governed by Snell's Law ($n_1\sin {\theta }_1=n_2\sin {\theta }_2$), resulting from a change in wave speed.
• Combining Waves: The principle of superposition leads to interference. Constructive interference requires a path difference of $n\lambda$; destructive interference requires $(n+\frac{1}{2})\lambda$. Diffraction is significant when a gap is similar in size to the wavelength.
• Stationary Waves: Standing waves with fixed nodes and antinodes form from interference of identical counter-propagating waves at resonant frequencies. The harmonic frequencies depend on the boundary conditions (fixed/fixed, open/open, or open/closed).