Free Pearson Edexcel International A Level Physics lesson

Wave energy and pulse echoes

Connect energy spreading, boundaries and useful image detail.

What you’ll learn

  • Calculate intensity from power and area.
  • Explain reflection and transmission at an interface.
  • Use round-trip travel time and pulse length to interpret echoes.

Power spread over an area

Intensity describes the rate at which energy crosses each square metre perpendicular to the direction of flow. A 2 W beam spread uniformly over 0.010 m² has intensity 200 W m⁻². This is different from energy, power or brightness judged by an eye. Quote the receiving area and its orientation; if the beam is nonuniform, total power divided by total area gives an average rather than the intensity at every point.

For an ideal source radiating uniformly in all directions, the same power crosses successive imaginary spheres. Their area is 4πr², so intensity falls as 1/r² if no energy is absorbed between spheres. Doubling the distance then reduces intensity to a quarter. This geometric model is unsuitable for an approximately collimated laser beam or a strongly directional speaker unless its spreading behaviour has separately been justified.

Absorption, reflection and changes in beam cross-section can all change the measured intensity. Keep these causes separate: a lower reading far away does not by itself prove that the medium absorbed the missing energy. In a fixed linear medium, intensity is proportional to displacement amplitude squared. Doubling amplitude can therefore quadruple intensity, so sound-level and laser safety cannot be judged from a modest-looking amplitude change alone.

I = P/A; for uniform spherical spreading I = P/(4πr²)

Symbols and units

  • I is intensity in W m⁻²; P is power in W; A is perpendicular area in m²; r is distance in m.

How the equation explains the idea

  1. Power distributed uniformly over area A gives I = P/A. For isotropic spreading without absorption, the same power crosses a sphere of area 4πr².
  2. Thus I = P/(4πr²). Doubling r spreads unchanged power over four times the area and reduces intensity to one quarter; it does not reduce the source power.
Doubling distance quarters intensity

Conditions. Isotropic point-source model with total power 4π W, no absorption, and distance from 1 m to 5 m. No claim is made at the singular source position r = 0.

Distance from source / m; Intensity / W m⁻².

I = 1/r² gives 1 W m⁻² at 1 m and 0.25 W m⁻² at 2 m. Power crossing each complete sphere remains 4π W despite the falling local intensity.

Open the interactive lesson to view the plotted curves and their coordinate tables.

Worked example

Spreading a detector beam

  1. A beam transfers 0.060 J in 0.020 s, so its power is 3.0 W.
  2. At the detector the uniform illuminated area is 0.015 m².
  3. Average intensity = 3.0/0.015 = 200 W m⁻².

Use power, not total energy, in I = P/A.

Keep this idea. Identify power and receiving area before calculating intensity or applying an inverse-square relation.

Reflection and transmission share an incoming wave

When a wave reaches an interface, the material response changes. Part of the disturbance may return into the first medium as a reflected wave while another part continues as a transmitted wave. Energy can also be absorbed. An interface does not have to choose only reflection or transmission. A pulse in a rope meeting a differently weighted section is a useful mechanical example, while ultrasound partly reflects at boundaries between tissues.

The source fixes the oscillation frequency for the reflected and transmitted periodic waves at a stationary boundary. If speed changes in the second medium, the transmitted wavelength changes in proportion to speed. A reflected pulse may be inverted or upright depending on the boundary conditions; do not assert that every reflected wave changes phase by 180°. The angle of reflection equals the angle of incidence when both are measured from the normal for a flat specular boundary.

Energy accounting is more reliable than adding displacement amplitudes blindly. In different media, the relation between amplitude and energy flux differs, so reflected and transmitted amplitude fractions need not add in the same way as power fractions. Ultrasound needs sufficient returned energy to detect a boundary but also enough transmitted energy to reach deeper boundaries. A coupling layer reduces the strong reflection that an air gap can otherwise cause at the probe surface.

Incident power = reflected power + transmitted power + absorbed power; f stays fixed at a stationary boundary.

Symbols and units

  • Incident, reflected, transmitted and absorbed powers are rates of energy transfer in W.
  • f is frequency in Hz, v is propagation speed in m s⁻¹ and λ is wavelength in m.

How the equation explains the idea

  1. Conservation of energy balances incoming power with outgoing plus absorbed power for a steady boundary with no energy accumulation.
  2. A stationary boundary preserves frequency. Since λ = v/f, a lower speed on the other side gives a proportionally shorter wavelength. Amplitude alone cannot determine transmitted power across different media.

Worked example

A boundary power budget

  1. An incoming wave carries 12 mW and a boundary reflects 3 mW.
  2. If absorption at that boundary is negligible, transmitted power = 12 − 3 = 9 mW.
  3. The reflected fraction is 3/12 = 0.25 and the transmitted fraction is 0.75.

Both outgoing paths are present; no extra energy is created.

Keep this idea. Track reflected, transmitted and absorbed energy separately; a stationary boundary preserves frequency.

Echo delay and detail are different questions

A pulse-echo instrument sends a short burst of waves and records returning signals against time. If the wave speed in a uniform medium is v and the pulse returns after delay Δt, a reflector directly along the beam is at distance vΔt/2. The factor of two accounts for the outgoing and returning paths. If the path crosses several materials, add travel times for the different path segments instead of using one unjustified speed.

Two close boundaries can produce overlapping echoes. A pulse of duration τ occupies a spatial length vτ; the separation in round-trip delays from boundaries a distance d apart is 2d/v. Distinct echoes require a delay difference sufficiently larger than the effective pulse duration and detector response. The simple estimate d ≳ vτ/2 describes axial pulse separation, not every contribution to image resolution.

Wavelength also limits the detail that wave scattering or diffraction can distinguish. Higher frequency in the same medium gives shorter wavelength and can reveal smaller structures, but attenuation may increase and reduce useful penetration. A shorter pulse and a shorter wavelength address related but distinct limitations. An apparently precise electronic clock cannot recover two boundaries whose echoes have merged into one broad signal, so judge the complete measurement system.

Depth = vΔt/2; pulse length = vτ; approximate axial separation scale = vτ/2

Symbols and units

  • v is wave speed in m s⁻¹; Δt is round-trip echo delay in s; τ is pulse duration in s.
  • Depth and pulse length are distances in m.

How the equation explains the idea

  1. The pulse travels to the reflector and back, so vΔt = 2 × depth. Divide by two only because the measured delay includes both legs.
  2. A pulse occupies length vτ. Two reflectors separated axially by d produce echo delays differing by 2d/v; requiring at least τ gives the approximate separation d ≥ vτ/2.

Worked example

A reflector and its neighbour

  1. A pulse returns after 80 μs in material where v = 1500 m s⁻¹.
  2. Depth = 1500 × 80 × 10⁻⁶/2 = 0.060 m.
  3. A 4.0 μs pulse gives an axial separation scale 1500 × 4.0 × 10⁻⁶/2 = 0.0030 m.

A 60 mm depth measurement does not imply that two boundaries 1 mm apart can be resolved.

Keep this idea. Use round-trip time for distance and assess pulse duration and wavelength separately for resolution.

Open the interactive lesson for free practice and reviewed three-question samples when available. New full question sets and timed unit checks are in preparation. Existing complete Physics revision packs retain paid access; existing free teaching PDFs remain free.

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