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 \(v = f\lambda\)

1. Speed definition – for any point of constant phase (e.g. a crest) \[ v = \frac{\text{distance travelled}}{\text{time taken}} = \frac{x_{2}-x_{1}}{t_{2}-t_{1}} . \]
2. Between two successive occurrences of the same phase the particle has moved one wavelength: \[ x_{2}-x_{1} = \lambda . \]
3. The time for one complete cycle to pass a fixed point is the period: \[ t_{2}-t_{1}=T . \]
4. Substituting (2) and (3) into (1) gives \[ v = \frac{\lambda}{T}. \]
5. 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

5. Intensity – Mechanical vs. Electromagnetic Waves

6. Measuring Frequency and Period with a Cathode‑Ray Oscilloscope (CRO)

1. Connect the signal source to the vertical input; set the horizontal sweep to a known time‑base (e.g. 1 ms/div).
2. Identify two successive identical points on the waveform (crest‑to‑crest or zero‑crossing). \[ T = (\text{number of divisions})\times(\text{time per division}). \]
3. Calculate the frequency: \(f = 1/T\).
4. 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 \(c = v\).)

1. Wavelength from the wave equation: \[ \lambda = \frac{v}{f}= \frac{340}{500}=0.68\ \text{m}. \]
2. RMS pressure amplitude: \[ p_{\text{rms}} = \frac{A}{\sqrt{2}} = 1.41\times10^{-5}\ \text{Pa}. \]
3. 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

1. If the frequency of a wave is doubled while the speed remains constant, what happens to the wavelength?
   Answer: \(\lambda\) is halved because \(\lambda = v/f\).
2. 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.
3. 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.