analyse and interpret graphical representations of transverse and longitudinal waves

Progressive Waves – Cambridge A‑Level Physics 9702 (Section 7.1)

In this section we study waves that, once created, travel through a uniform medium without any continued external driving. The aim is to analyse and interpret graphical representations of both transverse and longitudinal progressive waves and to apply the quantitative relationships that describe them.

1. Key Terms, Definitions & Formulae

Term	Definition / Symbol	Key Relation
Progressive (travelling) wave	A disturbance that moves through a medium, carrying energy from one point to another.	–
Transverse wave	Particle displacement ⟂ direction of propagation.	–
Longitudinal wave	Particle displacement ‖ direction of propagation.	–
Amplitude \(A\)	Maximum displacement from equilibrium (vertical for transverse, compression/rarefaction for longitudinal).	–
Wavelength \(λ\)	Distance between two successive points that are in the same phase (e.g. crest‑to‑crest or compression‑to‑compression).	–
Period \(T\)	Time for one complete cycle.	\(f = 1/T\)
Frequency \(f\)	Number of cycles per second (Hz).	\(f = 1/T\)
Phase difference \(Δφ\)	Angular separation between two points of the same wave. For a spatial separation \(Δx\):\(\displaystyle Δφ = kΔx = \frac{2π}{λ}\,Δx\).	–
Angular frequency \(ω\)	Rate of phase change in radians per second.	\(ω = 2πf\)
Wave‑number \(k\)	Spatial frequency of the wave.	\(k = 2π/λ\)
Phase speed \(v\)	Speed at which a point of constant phase travels.	\(v = fλ = ω/k\)

2. The Fundamental Wave Relation \(v = fλ\)

For any linear progressive wave in a uniform medium the phase speed is the product of its frequency and wavelength:

\[ v = fλ = \frac{ω}{k} \]

This relationship is used repeatedly in the exam. It can be derived from the sinusoidal wave equation (see Section 5) by noting that a fixed phase \((kx-ωt = \text{constant})\) moves a distance \(Δx\) in a time \(Δt\) such that \(kΔx = ωΔt\), giving \(Δx/Δt = ω/k = v\).

3. Intensity and Energy Transport

• Intensity \(I\) – average power transmitted per unit area perpendicular to the direction of propagation:
  \[ I = \frac{P}{A} \]
• For a sinusoidal wave in a linear medium the intensity is proportional to the square of the amplitude: \[ I \propto A^{2} \] Hence, doubling the amplitude increases the intensity by a factor of four.
• Energy density (energy per unit length of a string or per unit volume of a gas) is also proportional to \(A^{2}\). The phase speed \(v\) is independent of amplitude for linear media.
• Specific speed formulae:
  • String: \(\displaystyle v = \sqrt{\frac{T}{\mu}}\) (\(T\) = tension, \(\mu\) = linear mass density)
  • Sound in a fluid: \(\displaystyle v = \sqrt{\frac{B}{\rho}}\) (\(B\) = bulk modulus, \(\rho\) = density)

4. Mathematical Description of a Sinusoidal Progressive Wave

For a wave travelling in the +\(x\) direction:

\[ y(x,t)=A\sin\!\big(kx-\omega t+\phi\big) \]

• \(y\) – transverse displacement (for a string) or longitudinal quantity (e.g. particle displacement, pressure variation) for a sound wave.
• \(k\) – wave‑number, \(\displaystyle k = \frac{2π}{λ}\).
• \(ω\) – angular frequency, \(\displaystyle ω = 2πf\).
• \(\phi\) – initial phase (often taken as zero).

5. Graphical Representations Used in Exams

1. Displacement vs Position (snapshot at a fixed time) – shows the shape of the wave along the medium at a particular instant.
2. Displacement vs Time (time‑history at a fixed position) – shows the oscillation of a single particle as the wave passes.

Both graphs are sinusoidal for linear waves; the physical meaning of the vertical axis differs for transverse and longitudinal cases.

6. Interpreting a Displacement‑vs‑Position Graph

• Wavelength \(λ\): measured between two successive crests (transverse) or two successive compressions (longitudinal).
• Amplitude \(A\): vertical distance from the equilibrium line to a crest (or to the maximum compression/rarefaction).
• Direction of travel:
  • Pattern shifts to the right with increasing time → wave moves in the +\(x\) direction.
  • Pattern shifts to the left → wave moves in the –\(x\) direction.
• Local slope \(dy/dx\) (for longitudinal waves) is proportional to the strain in the medium and therefore to the pressure variation.
• Phase difference between two points separated by \(\Delta x\): \[ Δφ = kΔx = \frac{2π}{λ}\,Δx \] A shift of \(\pi\) corresponds to a half‑wavelength separation.

7. Interpreting a Displacement‑vs‑Time Graph

• Period \(T\): time between successive identical points (e.g. crest‑to‑crest). Frequency \(f = 1/T\).
• Amplitude \(A\): maximum vertical displacement from the equilibrium line.
• Phase information can be read by comparing the curve with a reference sinusoid (e.g. noting where it crosses the equilibrium line).
• For a travelling wave the same sinusoidal shape appears at different positions with a time lag: \[ Δt = \frac{Δx}{v} \] This relation is frequently used to combine a snapshot graph with a time‑history graph to obtain the wave speed.

8. Using a CRO (Cathode‑Ray Oscilloscope) to Determine Frequency & Amplitude

A CRO displays voltage (or displacement) on the vertical axis against time on the horizontal axis. In the exam you may be asked to interpret a CRO trace:

• Horizontal sweep speed \(S\) (units ms div\(^{-1}\)) is usually given. The time for one complete cycle is read from the width of one sinusoidal period on the screen: \(T = n\,S\) where \(n\) is the number of divisions.
• Vertical deflection \(V\) (or displacement) is read directly from the height of a crest. The amplitude is half the peak‑to‑peak value: \(A = V_{\text{peak}} = \frac{1}{2}V_{\text{p‑p}}\).
• Frequency follows from \(f = 1/T\). This method is common for sound‑wave experiments and for measuring the frequency of a string vibration.

9. Comparison of Transverse and Longitudinal Waves

Feature	Transverse Wave	Longitudinal Wave
Particle displacement direction	Perpendicular to propagation	Parallel to propagation
Typical examples	Light, string waves, water‑surface ripples	Sound in air, compression wave in a spring
Snapshot graph	Sinusoidal curve \(y\) vs \(x\)	Sinusoidal curve of compression/rarefaction (Δx or pressure) vs \(x\)
Time‑history graph	Vertical displacement \(y\) vs \(t\)	Pressure (or particle displacement) vs \(t\)
Energy transport	Kinetic + potential of the medium; intensity ∝ \(A^{2}\)	Pressure (potential) + particle motion (kinetic); intensity ∝ \(A^{2}\)
Speed in a uniform medium	\(v=\sqrt{T/μ}\)	\(v=\sqrt{B/ρ}\)

10. Worked Example – Determining Wave Speed from Two Graphs

Given:

• A snapshot graph of a transverse wave on a string shows a wavelength \(λ = 0.40\;\text{m}\) and an amplitude \(A = 2.0\;\text{mm}\).
• A time‑history graph taken at a fixed point on the same string shows a period \(T = 0.025\;\text{s}\).

Solution:

1. Read \(λ = 0.40\;\text{m}\) from the snapshot.
2. Period \(T = 0.025\;\text{s}\) ⇒ \(f = 1/T = 40\;\text{Hz}\).
3. Apply the wave equation \(v = fλ\): \[ v = 40\;\text{Hz}\times0.40\;\text{m}=16\;\text{m s}^{-1} \]

The wave travels at \(16\;\text{m s}^{-1}\) along the string.

11. Worked Example – Intensity Change with Amplitude

A sound wave in air has an amplitude three times larger than that of a reference wave. Both travel at the same speed. Find the ratio of their intensities.

Because \(I\propto A^{2}\):

\[ \frac{I_{\text{new}}}{I_{\text{ref}}}= \left(\frac{A_{\text{new}}}{A_{\text{ref}}}\right)^{2}=3^{2}=9 \]

The new wave is nine times more intense.

12. Typical Examination Tasks

• Read \(λ\) and \(A\) from a displacement‑vs‑position graph and state the direction of travel.
• Determine \(T\), \(f\) and \(A\) from a displacement‑vs‑time (or CRO) graph.
• Combine a snapshot graph with a time‑history graph to calculate the wave speed using \(v = λ/T\) or \(v = fλ\).
• Identify whether a given graph represents a transverse or longitudinal wave from the quantity plotted on the vertical axis.
• Calculate the phase difference between two points a known distance apart: \(Δφ = (2π/λ)Δx\).
• Quantify the change in intensity when the amplitude is altered (use \(I∝A^{2}\)).
• Explain how a CRO trace yields frequency and amplitude, and convert the screen divisions into physical values.

13. Key Points to Remember

1. The sinusoidal shape is identical for transverse and longitudinal waves; only the physical meaning of the vertical axis changes.
2. Wave speed is given by \(v = fλ\) and is independent of amplitude for linear media.
3. Energy density and intensity are proportional to the square of the amplitude (\(∝A^{2}\)), but this does not affect the speed.
4. Phase difference between two points: \(\displaystyle Δφ = kΔx = \frac{2π}{λ}Δx\). A shift of \(\pi\) corresponds to a half‑wavelength.
5. When interpreting graphs always check axis labels, units, and any indicated translation direction.
6. A progressive wave moves as a whole without changing shape; on a snapshot graph this appears as a constant horizontal translation over time.
7. Using a CRO: horizontal sweep gives the period, vertical deflection gives the amplitude, and \(f = 1/T\) follows directly.

14. Suggested Diagrams (placeholders for the examiner)