Beats: two notes throbbing
Sound two notes of nearly equal frequency together and you hear a slow throb — loud, soft, loud, soft. These are beats. Superposition explains them: the two waves drift into phase (loud) and out of phase (soft) over and over. The throbbing repeats at the beat frequency, equal to the difference of the two frequencies.
The beat frequency is the difference of the two source frequencies.
This is exactly how a musician tunes by ear. Play a 440\text{ Hz} tuning fork against a string at 443\text{ Hz} and you hear |443 - 440| = 3 beats per second. Adjust the string until the beats slow to nothing — now the frequencies match.
The Doppler effect
An ambulance races past and its siren drops in pitch the instant it passes. That is the Doppler effect: relative motion between source and observer shifts the observed frequency. As the source approaches, each new wavefront is emitted a little closer than the last, so the crests bunch up — shorter wavelength, higher pitch. As it recedes, the crests stretch out — longer wavelength, lower pitch.
Observed frequency f′; top signs for approach. v is the wave speed in the medium; v_o, v_s are observer and source speeds.
Read the signs so the shift goes the right way: motion that reduces the gap raises f', motion that opens the gap lowers it. Here v is the wave speed in the medium. For sound, a moving source and a moving observer are not symmetric — the air itself picks out a preferred frame. For light there is no medium, so only the relative velocity matters (the relativistic Doppler effect, a later story).
Worked example: a passing ambulance
Shock waves and where waves lead
Push the source speed all the way up to the wave speed and the wavefronts can no longer outrun it — they pile into a single high-pressure shock. At the speed of sound (Mach 1) an aircraft drags a cone-shaped shock that we hear as a sonic boom. It is the Doppler picture taken to its limit.
Here is the payoff. Everything in this track — v = f\lambda, superposition, interference, standing waves, resonance, the Doppler shift — carries over, unchanged in spirit, to light across the whole electromagnetic spectrum. Ultrasound imaging, radar speed guns, noise-canceling headphones, the diffraction of X-rays, and the redshift that tells us the universe is expanding are all this same physics. Even matter itself behaves as a wave in quantum mechanics.