Speed, frequency, wavelength, phase and energy transfer by waves
AQA A-Level Physics 3
📏Define amplitude, wavelength, frequency and period
⚡Use the wave equation v = fλ
🔄Define phase and calculate phase difference
💡Explain how intensity relates to amplitude: I ∝ A²
↔️Distinguish transverse and longitudinal waves
🌊Describe how energy is transferred by progressive waves
Wave Properties and Definitions
A progressive wave transfers energy from one place to another without transferring matter. The particles of the medium oscillate about their equilibrium positions.
Property
Symbol
Definition
SI unit
Amplitude
A
Maximum displacement from equilibrium
m
Wavelength
λ
Distance between adjacent points in phase (e.g. crest to crest)
m
Period
T
Time for one complete oscillation
s
Frequency
f
Number of complete oscillations per second
Hz
Wave speed
v
Speed at which the wave pattern moves
m s⁻¹
f = 1/T
v = fλ (wave equation)
Transverse wave: Oscillations are perpendicular to the direction of wave travel. Examples: light, water waves, S-waves.
Longitudinal wave: Oscillations are parallel to the direction of wave travel. Examples: sound, P-waves. They have compressions and rarefactions.
Phase and Phase Difference
Phase: The stage of an oscillation cycle that a point has reached. Points at the same phase oscillate identically at the same time.
Phase difference: The difference in phase between two points on a wave or on two different waves. Measured in radians or degrees, or as a fraction of a wavelength.
Phase difference φ = (2π × x) / λ (in radians)
or: φ = (x / λ) × 360° (in degrees)
where x = path difference (m)
Path difference (x)
Phase difference (rad)
Phase relationship
0, λ, 2λ...
0, 2π, 4π...
In phase
λ/2, 3λ/2...
π, 3π...
Antiphase (180° out of phase)
λ/4
π/2
90° out of phase
Two points separated by exactly one wavelength are in phase (phase difference = 2π rad = 360°). Two points separated by half a wavelength are in antiphase (π rad = 180°).
Intensity and Amplitude
Intensity: The power transmitted per unit area perpendicular to the direction of wave travel. SI unit: W m⁻².
I = P / A (where A is area, not amplitude)
For a point source (spherical spreading): I = P / (4πr²)
Key relationship: I ∝ A² (intensity ∝ amplitude squared)
Since I ∝ A² and I ∝ 1/r² for a point source, we can write:
A ∝ 1/r (amplitude decreases with distance for a point source)
Doubling the amplitude quadruples the intensity. Halving the amplitude reduces intensity to one quarter. This is a fundamental relationship for all wave types.
Do not confuse amplitude (A, metres) with area (also A, metres²) or with the constant in I = P/A. The context makes the meaning clear.
Energy Transfer by Waves
Progressive waves transfer energy from source to receiver without transferring matter. The oscillating particles of the medium pass energy along the wave by doing work on adjacent particles.
Mechanical waves require a medium (e.g. sound, water waves, seismic waves). The medium oscillates but does not travel with the wave.
Electromagnetic waves do not require a medium — they can travel through a vacuum. They consist of oscillating electric and magnetic fields perpendicular to each other and to the direction of travel. All EM waves travel at c = 3.00 × 10⁸ m s⁻¹ in a vacuum.
The EM spectrum (in order of increasing frequency): radio, microwave, infrared, visible (red→violet), UV, X-rays, gamma rays. All travel at c in vacuum but have different frequencies, wavelengths and photon energies.
For electromagnetic waves in a vacuum: v = c = fλ → λ = c/f. For sound in air at 20°C: v ≈ 340 m s⁻¹.
A sound wave has frequency 440 Hz and travels at 340 m s⁻¹. Calculate its wavelength and period.
1v = fλ → λ = v/f = 340 / 440 = 0.773 m
2T = 1/f = 1/440 = 2.27 × 10⁻³ s
λ = 0.773 m (77.3 cm); T = 2.27 × 10⁻³ s = 2.27 ms
Two points on a wave are 0.30 m apart. The wavelength is 0.80 m. Calculate the phase difference in degrees and radians.
A loudspeaker produces a sound wave with intensity 4.0 × 10⁻³ W m⁻² at 2.0 m from the source. Calculate (a) the power output and (b) the intensity at 6.0 m.
1I = P/(4πr²) → P = I × 4πr² = 4.0 × 10⁻³ × 4π × (2.0)² = 4.0 × 10⁻³ × 50.27 = 0.201 W
2At r = 6.0 m: I = P/(4πr²) = 0.201 / (4π × 36) = 0.201 / 452.4 = 4.44 × 10⁻⁴ W m⁻²
A wave has amplitude 3.0 cm. Its intensity is 18 W m⁻². What is the intensity if the amplitude is increased to 9.0 cm?
1I ∝ A² → I₂/I₁ = (A₂/A₁)² = (9.0/3.0)² = 3² = 9
2I₂ = 9 × 18 = 162 W m⁻²
I₂ = 162 W m⁻²
1. Which of the following is a longitudinal wave?
Sound waves are longitudinal — particles oscillate parallel to the direction of travel (compressions and rarefactions). All EM waves (light, microwaves) are transverse.
2. A wave has wavelength 2.4 m and frequency 5.0 Hz. What is its speed?
v = fλ = 5.0 × 2.4 = 12 m s⁻¹
3. Two points are separated by 3λ/4. What is their phase difference in radians?
φ = (2π/λ) × x = (2π/λ) × (3λ/4) = 3π/2 rad = 270°.
4. Intensity is doubled. By what factor does the amplitude change?
I ∝ A² → A ∝ √I. If I doubles, A increases by factor √2 ≈ 1.41.
5. Radio waves have frequency 100 MHz. Calculate their wavelength. (c = 3.00 × 10⁸ m s⁻¹)
1. A point source emits sound uniformly in all directions. At 3.0 m, the intensity is 8.0 × 10⁻³ W m⁻². (a) Find the power of the source. (b) Find the intensity at 12.0 m. (c) Find the amplitude ratio A(12m)/A(3m).
(a) P = I × 4πr² = 8.0×10⁻³ × 4π × 9 = 9.047×10⁻¹ ≈ 0.905 W. (b) I ∝ 1/r². Ratio (r₂/r₁)² = (12/3)² = 16. I₂ = 8.0×10⁻³/16 = 5.0×10⁻⁴ W m⁻². (c) A ∝ 1/r (since I ∝ A² and I ∝ 1/r²). A(12)/A(3) = 3/12 = 1/4 = 0.25.
2. Ultrasound of frequency 5.0 MHz is used in medical imaging. The speed of ultrasound in soft tissue is 1540 m s⁻¹. Calculate the wavelength and explain why high frequency is preferred for imaging.
λ = v/f = 1540 / (5.0×10⁶) = 3.08×10⁻⁴ m = 0.308 mm. Higher frequency → shorter wavelength → better resolution (can resolve smaller structures). Lower frequency ultrasound has longer wavelengths and would not resolve fine tissue details. However, higher frequency ultrasound is attenuated more rapidly in tissue, limiting penetration depth — there is a trade-off between resolution and penetration.
3. Explain the difference between the speed of a wave and the speed of the oscillating particles in the medium. Using the equation v = fλ, explain what determines the wave speed in a medium.
Wave speed v is the speed at which the wave pattern (energy) travels through the medium — it depends on the properties of the medium (density, elasticity), not on the source. Particle speed is the speed at which the medium particles oscillate about their equilibrium positions — it depends on frequency and amplitude and varies during each cycle (maximum at equilibrium, zero at maximum displacement). For a given medium, v is constant regardless of frequency. So v = fλ means that if frequency increases, wavelength must decrease proportionally to maintain the same wave speed in that medium. For example, sound travels at ~340 m s⁻¹ in air regardless of pitch.