Understanding Waves
Waves carry energy without carrying matter. This guide builds from basic wave properties through the pendulum, Doppler effect, interference, and sound intensity — with the intuition behind every formula.
All waves — light, sound, water — obey the same fundamental relationships between speed, frequency and wavelength. Master these and you can analyse any wave.
The speed of a wave is determined by the medium it travels through — the stiffness and density of the material. The source only determines how often it vibrates (frequency). Once the wave is in the medium, the medium controls how fast that disturbance propagates. Change the source and you change the wavelength, not the speed.
b) A wave has λ = 0.4 m and v = 320 m/s. Find f and T.
c) When a wave enters a denser medium and slows down, does the frequency change? Why?
a) λ = v/f = 340/200 = 1.7 m
b) f = v/λ = 320/0.4 = 800 Hz T = 1/f = 1/800 = 0.00125 s
c) No. Frequency is set by the source and is unchanged as the wave crosses into a new medium. Only wavelength changes (it shrinks as v decreases, since f stays constant and v = fλ).
The pendulum is the classic example of Simple Harmonic Motion (SHM). Its period depends only on length and gravity — not on mass or how far it swings (for small angles). This property is called isochronism and is why pendulum clocks work.
Swing wider and the bob travels a longer arc. But it also moves faster (gravity pulls harder from further out). These two effects exactly cancel for small angles: more distance, more speed — same time. This is a deep property of any system where restoring force is proportional to displacement.
A heavier bob experiences more gravitational force, but it also has more inertia — harder to accelerate. The two effects cancel exactly. This is the same reason all objects fall at the same rate in a vacuum: gravity scales with mass, and so does inertia.
b) Same pendulum on the Moon (g = 1.62 m/s²). Find T.
c) Why does doubling the mass of a pendulum bob not change the period?
a) T = 2π√(0.5/9.8) = 2π√(0.051) = 2π × 0.226 = 1.42 s
b) T = 2π√(0.5/1.62) = 2π√(0.309) = 2π × 0.556 = 3.49 s (much slower on the Moon)
c) Mass cancels out. Gravity pulls harder on a heavier bob, but that same mass has more inertia and resists acceleration equally. The T = 2π√(L/g) formula contains no mass term — they cancel algebraically.
When a source of waves and an observer are moving relative to each other, the observed frequency differs from the emitted frequency. The actual wave speed through the medium does not change.
As the source moves toward you, each successive wavefront is emitted from a position closer to you. The wavefronts pile up — they arrive more frequently. More wavefronts per second means higher observed frequency. When the source recedes, wavefronts spread apart and arrive less frequently — lower pitch.
b) Same train receding at 40 m/s. What frequency does the observer hear?
c) Does the Doppler effect change the speed of sound reaching the observer?
a) fobs = 500 × 340/(340−40) = 500 × 340/300 = 567 Hz
b) fobs = 500 × 340/(340+40) = 500 × 340/380 = 447 Hz
c) No. The speed of sound in air is determined by the medium (~340 m/s at 20°C) and is unaffected by source or observer motion. Only the observed frequency changes.
When two waves overlap in the same medium, their displacements add (superposition). If crests meet crests, the result is louder (constructive). If a crest meets a trough, they cancel (destructive). Standing waves form when a wave reflects back on itself, creating fixed nodes and antinodes.
At a node, the forward-travelling wave and the reflected wave always arrive with exactly opposite displacements — they cancel at every instant. It's permanent destructive interference at that point. Between nodes, the wave oscillates up and down (antinode region) but the nodal points themselves never move.
path diff = (0 + ½)λ → this matches the destructive condition with n = 0.
b) Two waves have a path difference of 2.5λ. Is the interference constructive or destructive?
c) Where are the nodes on a standing wave, and why don't they move?
a) fn = nv/(2L): f1 = 320/(1.6) = 200 Hz f2 = 400 Hz f3 = 600 Hz
b) 2.5λ = (2 + ½)λ, so n=2 → this is the destructive pattern. Destructive interference (quiet point).
c) Nodes sit at positions where the forward and reflected waves always arrive with opposite phase — they permanently destructively interfere. No matter what instant you look, the displacements cancel exactly at those points, so they never move.
Sound intensity (I) is the power per unit area (W/m²). As sound spreads outward spherically from a point source, the same power is spread over an ever-larger sphere. The surface area of a sphere grows as r², so intensity falls as 1/r².
Imagine the energy spreading over a sphere of radius r. Area = 4πr². If you double r, the area quadruples, so the energy per unit area (intensity) drops to one quarter. Move three times as far away and intensity drops to 1/9. This geometric spreading is why sound fades so quickly with distance.
Our ears respond logarithmically. A sound 10 times more intense feels roughly twice as loud (not 10 times). The decibel scale compresses this enormous range (10&sup-;¹² to 10² W/m²) into a manageable 0–140 dB scale that reflects how we actually perceive loudness.
b) The intensity of a sound triples. By how many dB does the level increase? (Hint: 10 log 3 ≈ 4.77)
c) Why does moving twice as far from a speaker make it sound much quieter than expected?
a) L = 10 log(10&sup-;&sup8;/10&sup-;¹²) = 10 log(10&sup4;) = 10 × 4 = 40 dB
b) ΔL = 10 log(3I/I) = 10 log(3) ≈ 10 × 0.477 = 4.77 dB
c) Because of the inverse square law. Doubling the distance reduces intensity to 1/4 (not 1/2). On the dB scale that's a drop of 10 log(1/4) = −6 dB. Our ears perceive this as noticeably quieter — the drop is steeper than intuition suggests because intensity falls with the square of distance.