derive, using the definitions of speed, frequency and wavelength, the wave equation v = f λ
Progressive Waves – Cambridge International AS & A Level Physics (9702)
Learning Objective
Derive, using the definitions of speed, frequency and wavelength, the wave equation
\[
v = f\lambda
\]
and relate the other fundamental wave parameters (displacement, amplitude, phase‑difference, period, intensity) to the behaviour of a progressive wave.
1. Wave Parameters – Definitions
| Parameter | Symbol | Definition | Typical Units |
|---|---|---|---|
| Displacement | \(y\) | Instantaneous distance of a particle from its equilibrium position. | m |
| Amplitude | A | Maximum displacement of a particle from equilibrium (peak of the wave). | m |
| Phase‑difference | \(\Delta\phi\) | Angular separation between two points of a wave; \(\displaystyle\Delta\phi = 2\pi\frac{\Delta x}{\lambda}=2\pi f\Delta t\). | rad |
| Period | T | Time for one complete cycle to pass a fixed point. | s |
| Frequency | f | Number of cycles per second; \(\displaystyle f = \frac{1}{T}\). | Hz |
| Wavelength | \(\lambda\) | Spatial distance between two consecutive points that are in phase (e.g. crest‑to‑crest). | m |
| Wave speed | v | Distance a given phase travels per unit time. | m s⁻¹ |
| Intensity | I | Power transmitted per unit area normal to the direction of propagation. | W m⁻² |
2. Visualising a Progressive Wave
3. Derivation of the Wave Equation \(v = f\lambda\)
- Speed definition – for any point of constant phase (e.g. a crest)
\[
v = \frac{\text{distance travelled}}{\text{time taken}} = \frac{x{2}-x{1}}{t{2}-t{1}} .
\]
- Between two successive occurrences of the same phase the particle has moved one wavelength:
\[
x{2}-x{1} = \lambda .
\]
- The time for one complete cycle to pass a fixed point is the period:
\[
t{2}-t{1}=T .
\]
- Substituting (2) and (3) into (1) gives
\[
v = \frac{\lambda}{T}.
\]
- Because frequency is the reciprocal of the period, \(f = 1/T\), we have \(T = 1/f\). Re‑placing \(T\) yields
\[
v = \frac{\lambda}{1/f}=f\lambda .
\]
Thus, for any progressive wave,
\[
\boxed{v = f\lambda}
\]
4. Wave‑Speed Formulas for Common Media
- String (transverse wave) – tension \(T\) and linear mass density \(\mu\):
\[
v = \sqrt{\frac{T}{\mu}} .
\]
- Sound in a fluid (longitudinal wave) – bulk modulus \(B\) and density \(\rho\):
\[
v = \sqrt{\frac{B}{\rho}} .
\]
- Electromagnetic wave in vacuum:
\[
v = c = 3.00\times10^{8}\ \text{m s}^{-1}.
\]
- EM wave in a dielectric – relative permittivity \(\varepsilon_{r}\):
\[
v = \frac{c}{\sqrt{\varepsilon_{r}}}.
\]
5. Intensity – Mechanical vs. Electromagnetic Waves
Mechanical (e.g. string, sound)
- Average power transmitted by a harmonic wave on a string:
\[
P_{\text{avg}} = \tfrac12\,\mu\,\omega^{2}A^{2}v ,
\qquad \omega = 2\pi f .
\]
- Average intensity (power per unit area perpendicular to propagation):
\[
I = \frac{P{\text{avg}}}{A{\perp}} \propto A^{2}.
\]
Electromagnetic (e.g. light, radio)
- For a plane wave in vacuum the time‑averaged intensity is
\[
I = \frac{1}{2}\,c\,\varepsilon{0}\,E{0}^{2}
= \frac{1}{2}\,\frac{c}{\mu{0}}\,B{0}^{2},
\]
where \(E{0}\) and \(B{0}\) are the peak electric and magnetic field amplitudes.
- Again, \(I\propto E{0}^{2}\propto B{0}^{2}\), so the same \(I\propto A^{2}\) proportionality holds for EM waves.
6. Measuring Frequency and Period with a Cathode‑Ray Oscilloscope (CRO)
- Connect the signal source to the vertical input; set the horizontal sweep to a known time‑base (e.g. 1 ms/div).
- Identify two successive identical points on the waveform (crest‑to‑crest or zero‑crossing).
\[
T = (\text{number of divisions})\times(\text{time per division}).
\]
- Calculate the frequency: \(f = 1/T\).
- If the wave propagates in a medium whose speed \(v\) is known (from the formulas in §4), obtain the wavelength from \( \lambda = v/f\).
7. Worked Example – Sound Wave in Air
Problem: A sound wave travels in air at \(v = 340\ \text{m s}^{-1}\) and has a frequency of \(f = 500\ \text{Hz}\). Find its wavelength and intensity, given a pressure‑amplitude \(A = 2.0\times10^{-5}\ \text{Pa}\). (For sound in air, \(I = p_{\text{rms}}^{2}/(\rho c)\) with \(\rho = 1.2\ \text{kg m}^{-3}\) and \(c = v\).)
- Wavelength from the wave equation:
\[
\lambda = \frac{v}{f}= \frac{340}{500}=0.68\ \text{m}.
\]
- RMS pressure amplitude:
\[
p_{\text{rms}} = \frac{A}{\sqrt{2}} = 1.41\times10^{-5}\ \text{Pa}.
\]
- Intensity:
\[
I = \frac{p_{\text{rms}}^{2}}{\rho c}
= \frac{(1.41\times10^{-5})^{2}}{1.2 \times 340}
\approx 4.9\times10^{-13}\ \text{W m}^{-2}.
\]
Result: \(\lambda = 0.68\ \text{m}\), \(I \approx 5\times10^{-13}\ \text{W m}^{-2}\).
8. Conceptual Check Questions
- If the frequency of a wave is doubled while the speed remains constant, what happens to the wavelength?
Answer: \(\lambda\) is halved because \(\lambda = v/f\).
- Explain why the wave equation \(v = f\lambda\) holds for both transverse and longitudinal waves.
Answer: The equation follows solely from the definitions of speed (distance / time), frequency (cycles / time) and wavelength (distance per cycle). Whether the particle motion is perpendicular (transverse) or parallel (longitudinal) to the direction of travel, one complete cycle still travels a distance \(\lambda\) in a time \(T = 1/f\), giving the same relationship.
- How does the intensity of a progressive wave change if the amplitude is increased by a factor of 3?
Answer: Since \(I \propto A^{2}\), the intensity increases by a factor of \(3^{2}=9\).
9. Connection to Subsequent Wave Topics
The parameters introduced here—\(v, f, \lambda, A, I\) and phase—are reused throughout the wave chapter. In the next sections you will:
- Apply the superposition principle to analyse interference and standing waves (Sections 7.2‑7.3).
- Use the Doppler‑effect formulas, which rely on the same \(v = f\lambda\) relationship.
- Explore electromagnetic wave phenomena such as reflection, refraction and polarisation (Section 7.5), where the intensity expressions for EM waves become essential.
- Combine the speed formulas of §4 with the wave equation to solve mixed‑media problems (e.g., light entering glass).
10. Summary
- All fundamental wave parameters are inter‑related; the universal wave equation \(v = f\lambda\) follows directly from the definitions of speed, frequency and wavelength.
- Wave speed depends on the medium: strings (\(\sqrt{T/\mu}\)), fluids (\(\sqrt{B/\rho}\)), vacuum (\(c\)).
- Intensity is proportional to the square of the amplitude for both mechanical and electromagnetic waves, with the full textbook expressions shown in §5.
- Practical measurement of \(f\) and \(T\) can be performed with a CRO; wavelength follows from the wave equation once \(v\) is known.
- Mastering these relationships provides the foundation for later topics such as interference, diffraction, the Doppler effect and polarisation, all of which are part of the Cambridge AS & A Level syllabus.