analyse and interpret graphical representations of transverse and longitudinal waves

Progressive Waves – Cambridge A‑Level Physics 9702

1️⃣ What is a Progressive Wave?

A progressive wave is a disturbance that travels through a medium, carrying energy from one place to another without transporting matter. Think of a ripple moving across a pond when you drop a stone – the water itself doesn’t move far, but the wave does.

2️⃣ Transverse vs. Longitudinal Waves

  • 🔵 Transverse wave – particle displacement is perpendicular to the direction of propagation. Example: waves on a string or electromagnetic waves.
  • 🔴 Longitudinal wave – particle displacement is parallel to the direction of propagation. Example: sound waves in air.

3️⃣ Graphical Representation

On a graph, the horizontal axis usually represents position (x) or time (t), while the vertical axis shows the displacement (y) or pressure (p).

Transverse Wave Example

$$y(x,t)=A\sin(kx-\omega t)$$

  • $A$ – amplitude (maximum displacement)
  • $k=\frac{2\pi}{\lambda}$ – wave number (related to wavelength $\lambda$)
  • $\omega=2\pi f$ – angular frequency (related to frequency $f$)

Longitudinal Wave Example

$$p(x,t)=p_0 + \Delta p \sin(kx-\omega t)$$

  • $p_0$ – ambient pressure
  • $\Delta p$ – pressure amplitude

4️⃣ Key Relationships

Quantity Formula
Wave speed $$v = f\lambda = \frac{\omega}{k}$$
Frequency–period relation $$f = \frac{1}{T}$$
Energy per unit volume (transverse) $$E = \frac{1}{2}\mu \omega^2 A^2$$

5️⃣ Exam Tips Box

🔍 Remember:

  • Identify whether the wave is transverse or longitudinal from the diagram.
  • Use the correct formula for wave speed: $v = f\lambda$ or $v = \frac{\omega}{k}$.
  • Check units – metres per second (m/s) for speed, hertz (Hz) for frequency.
  • When asked for amplitude, look for the maximum displacement from the equilibrium line.
  • For energy questions, use the appropriate energy density expression.

6️⃣ Practice Questions

  1. On a graph of a transverse wave, the distance between two successive peaks is 0.5 m and the wave travels at 4 m s⁻¹. What is the frequency?
  2. A longitudinal sound wave in air has a wavelength of 0.68 m. If the speed of sound is 340 m s⁻¹, calculate the frequency.
  3. Given the wave equation $y(x,t)=0.02\sin(4\pi x-6\pi t)$, find the amplitude, wavelength, and period.

7️⃣ Analogy Corner 🌟

Think of a transverse wave like a seesaw – the up and down motion is perpendicular to the line of the seesaw. A longitudinal wave is like a crowd wave at a stadium – people push and pull in the same direction as the wave moves.

Revision

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