Wave Speed Basics
Most people learn v = f in physics class and think that's the whole answer. It isn't. That formula tells you the relationship between speed, frequency, and wavelength for a periodic wave in a uniform medium. It does not tell you how to measure speed when you're standing in a lab with a rope and a function generator, or how to handle situations where the medium changes properties partway through the wave's path. The direct method depends entirely on what kind of wave you're dealing with and what equipment you have available. For mechanical waves — things on strings, in pipes, on springs — you can measure it physically. Set up a standing wave, count the nodes, measure the distance between them to get wavelength, then multiply by the driving frequency. That gives you wave speed. I've done this with a vibrato arm attached to a meter stick and a frequency counter for years. It works fine until the tension isn't uniform along the string, which happens more often than people expect. For electromagnetic waves in a vacuum, the speed is c, approximately 3 × 10^8 m/s. That's a constant. In other media, you divide c by the refractive index of the material. Glass around 1.5 means light travels at roughly 2 × 10^8 m/s through it. This is straightforward until you're working with dispersive materials where the refractive index changes with frequency, and then your single-number answer becomes a range depending on what wavelength you're actually measuring.
Sound waves in air are different again. The speed depends on temperature, humidity, and to a lesser extent pressure. At 20°C in dry air, sound moves at about 343 m/s. The formula v = 331 + 0.6T gives you a reasonable estimate where T is temperature in Celsius. I measured sound speed once in a long hallway using a balloon pop and two microphones connected to an oscilloscope. The distance was 12.4 meters. The time delay between the two mics was 36.1 milliseconds. That gave me 343.5 m/s. The walls were unfinished drywall and the HVAC was running, which introduced enough background noise that I had to average six measurements to get a stable reading. When you can't set up a standing wave or use timing equipment, there are indirect approaches. For ocean waves, you can track individual wave crests with a stopwatch and a known reference distance. For seismic waves, you use the time difference between P-wave and S-wave arrival at a seismograph station. The farther the station is from the epicenter, the larger that time gap becomes, and the relationship between that gap and distance is well mapped out. One thing beginners consistently miss is that wave speed and particle speed are not the same thing. In a transverse wave on a string, the wave travels down the string at some speed while individual points on the string move up and down. Confusing these two leads to incorrect answers on problems involving energy transfer and intensity calculations. The wave speed is determined by the medium's properties — tension and linear mass density for a string. The particle speed depends on the amplitude and frequency of the oscillation.
Another counter-intuitive point: when a wave moves from one medium to another, its speed changes, its wavelength changes, but its frequency stays the same. People sometimes think the frequency shifts because they intuitively expect something to change across the boundary. It doesn't. The boundary conditions force the oscillation rate to be continuous across the interface. Here's a practical problem I ran into last year. I was measuring wave speed on a heavy nylon rope under high tension, and the calculated speed from v = (T/) didn't match the measured speed from the standing wave pattern. The discrepancy was about eight percent. I spent two days checking tension measurements, re-measuring the linear mass density, and ruling out slippage at the clamps. The real issue turned out to be stiffness in the rope. The standard formula assumes a perfectly flexible string, but a thick nylon rope has bending rigidity that becomes significant at higher frequencies. The effective wave speed increased slightly because the stiffness added restoring force beyond what tension alone provided. I corrected it by using the more complete equation that includes a term for flexural rigidity, which involved measuring the rope's Young's modulus and cross-sectional geometry. It took about an afternoon once I knew what I was looking for, but the standard textbook approach would have left me with an unexplained error margin. There are situations where all the standard methods break down. In highly dispersive media, different frequency components travel at different speeds, so a single wave speed doesn't really exist for a complex signal. Pulse distortion becomes significant over distance. In nonlinear media, the wave speed can depend on amplitude itself, which means a loud sound travels differently than a quiet one. These are edge cases for most practical work, but they matter if you're doing anything beyond introductory physics problems.
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If you need a quick reference for common wave speeds, air at room temperature for sound is 343 m/s, steel for longitudinal waves is roughly 5900 m/s, guitar strings typically range from 100 to 400 m/s depending on tension and gauge, and light in water is about 2.25 × 10^8 m/s. These numbers are useful for sanity-checking your calculations.
