During one period \(T\) a progressive sinusoidal wave advances one wavelength \(\lambda\). By definition of speed,
\[
v = \frac{\lambda}{T}.
\]
Since \(f = \dfrac{1}{T}\), we obtain the universally applicable relation
\[
\boxed{v = f\lambda }.
\]
This holds for any progressive wave—transverse or longitudinal—provided the medium is uniform.
| Medium | Wave‑speed expression | Typical variables |
|---|---|---|
| String under tension | \(v = \sqrt{\dfrac{T}{\mu}}\) | \(T\) = tension (N), \(\mu\) = linear mass density (kg m⁻¹) |
| Longitudinal wave in a thin rod | \(v = \sqrt{\dfrac{E}{\rho}}\) | \(E\) = Young’s modulus (Pa), \(\rho\) = density (kg m⁻³) |
| Sound in a gas | \(v = \sqrt{\dfrac{\gamma\,p}{\rho}}\) | \(\gamma\) = ratio of specific heats, \(p\) = pressure (Pa), \(\rho\) = density (kg m⁻³) |
| Electromagnetic wave in vacuum | \(c = 3.00\times10^{8}\ \text{m s}^{-1}\) | independent of \(f\) and \(\lambda\) |
\[
v = f\lambda = (500)(0.68)=340\ \text{m s}^{-1}.
\]
\[
f = \frac{v}{\lambda}= \frac{12}{0.30}=40\ \text{Hz}.
\]
\[
f = \frac{1}{T}=40\ \text{Hz},\qquad
\lambda = \frac{v}{f}= \frac{5}{40}=0.125\ \text{m}.
\]
\[
v = \sqrt{\frac{T}{\mu}}=\sqrt{\frac{80}{0.010}}=89.4\ \text{m s}^{-1}.
\]
If the measured wavelength is \(0.45\ \text{m}\), then \(f = v/\lambda = 199\ \text{Hz}\).
When source and/or observer move relative to the medium, the observed frequency \(f'\) differs from the emitted frequency \(f\). For sound in a stationary medium:
\[
f' = f\,\frac{v \pm v{\text{obs}}}{v \pm v{\text{src}}}.
\]
Worked example (ambulance siren):
Source frequency \(f = 800\ \text{Hz}\); source speed \(v_{\text{src}} = 30\ \text{m s}^{-1}\) towards a stationary observer; speed of sound \(v = 340\ \text{m s}^{-1}\).
\[
f' = 800\;\frac{340}{340-30}= 800\;\frac{340}{310}= 878\ \text{Hz}.
\]
| Region | Typical Wavelength \(\lambda\) | Frequency \(f\) (Hz) | Common Example |
|---|---|---|---|
| Radio | \(>10^{3}\ \text{m}\) | \(<10^{6}\) | Broadcasting |
| Microwave | \(10^{-2}–10^{0}\ \text{m}\) | \(10^{9}–10^{11}\) | Radar |
| Infrared | \(7\times10^{-7}–10^{-3}\ \text{m}\) | \(3\times10^{11}–4\times10^{14}\) | Thermal imaging |
| Visible | \(4\times10^{-7}–7\times10^{-7}\ \text{m}\) | \(4.3\times10^{14}–7.5\times10^{14}\) | Sunlight |
| Ultraviolet | \(10^{-8}–4\times10^{-7}\ \text{m}\) | \(7.5\times10^{14}–3\times10^{16}\) | Sunburn |
| X‑ray | \(10^{-11}–10^{-8}\ \text{m}\) | \(3\times10^{16}–3\times10^{19}\) | Medical imaging |
| Gamma | \(<10^{-11}\ \text{m}\) | \(>3\times10^{19}\) | Radioactive decay |
All EM waves travel at \(c = f\lambda\) in vacuum; the relationship is also valid in any transparent medium with the appropriate refractive index.
Only transverse waves can be polarised. Linear polarisation restricts the electric‑field vector to a single plane.
\[
I = I_{0}\cos^{2}\theta,
\]
where \(\theta\) is the angle between the incident polarisation direction and the transmission axis of the analyser.
\[
P_{\text{avg}} = \frac{1}{2}\,\mu A^{2} v \,\omega^{2}
= \frac{1}{2}\,\mu A^{2} v\,(2\pi f)^{2}.
\]
\[
I = \frac{P{\text{avg}}}{A{\text{cross}}}.
\]
\[
I \propto A^{2}.
\]
A CRO plots voltage versus time, allowing direct reading of wave parameters.
| Wave Type | Particle Motion | Typical Example | Speed Formula (AO2) |
|---|---|---|---|
| Transverse | Perpendicular to propagation | Wave on a string, surface water wave | \(v = \sqrt{T/\mu}\) (string) |
| Longitudinal | Parallel to propagation | Sound in air, longitudinal wave in a steel rod | \(v = \sqrt{E/\rho}\) (rod) or \(v = \sqrt{\gamma p/\rho}\) (gas) |
The resultant displacement at any point is the algebraic sum of the individual displacements:
\[
y{\text{total}} = y{1}+y_{2}+ \dots
\]
This leads directly to constructive (\(y{\text{total}}\) maximum) and destructive (\(y{\text{total}}=0\)) interference.
Two coherent sources separated by distance \(d\) produce an interference pattern on a screen at distance \(D\). For small angles (\(\sin\theta \approx \tan\theta = y/D\)) the condition for bright (constructive) fringes is
\[
d\sin\theta = n\lambda \quad (n = 0, \pm1, \pm2,\dots).
\]
Thus the fringe spacing \(\Delta y\) is
\[
\Delta y = \frac{\lambda D}{d}.
\]
Example: \(d = 0.30\ \text{mm}\), \(D = 2.0\ \text{m}\), \(\lambda = 600\ \text{nm}\).
\[
\Delta y = \frac{6.0\times10^{-7}\times2.0}{3.0\times10^{-4}} = 4.0\ \text{mm}.
\]
For a slit of width \(a\), the first minimum occurs when
\[
a\sin\theta = \lambda.
\]
When the slit width is comparable to \(\lambda\), a central bright band with diminishing side bands is observed.
Boundary conditions restrict the allowed wavelengths on a fixed‑fixed string:
\[
L = \frac{n\lambda}{2}\quad (n = 1,2,3,\dots).
\]
Measuring node‑antinode spacing (\(\lambda/2\)) allows the experimental determination of \(v\) via \(v = f\lambda\).
In a non‑uniform medium the wave speed varies with frequency, so the simple relation \(v = f\lambda\) still holds locally but \(v\) is no longer a constant for all components of a wave packet. Typical A‑Level examples:
In any uniform medium the three core quantities of a progressive wave are linked by the simple, universally applicable equation
\[
\boxed{v = f\lambda }.
\]
Knowing any two of \(v\), \(f\) and \(\lambda\) allows the third to be found. Combined with the appropriate speed formula for the medium, this relationship underpins the calculation of wave speed, the analysis of standing waves, and the interpretation of Doppler, interference and diffraction phenomena. Mastery of these concepts, the associated measurement techniques (CRO, Polaroid, double‑slit), and an awareness of limits such as dispersion, are essential for success in the Cambridge A‑Level Physics 9702 examinations.
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