Wave Speed

G14-2. Wave Speed and What It Depends On

This short guide addresses a misconception about wave speed and what it depends on. We'll start with some examples of factors that influence the wave speed in various situations.

Waves in strings and springs: The wave speed in a stretched string or spring depends on two factors, the tension F_T in the medium and the linear density μ (mass per unit length) of the medium. The equation relating these is v = (F_T/μ)^0.5.

Waves in air: The wave speed in dry air depends on the pressure P and density ρ of the air according to this relationship: v = c(P/ρ)^0.5, where c is a constant.

Note that in both of these cases, the functional form of the relationship is the same. The wave speed depends on the square root of the ratio of two quantities. One of the quantities is elastic (the numerator) and the other is inertial (the denominator). In general, the speed of a wave in a medium is given by v = (elastic property/inertial property)^0.5. An increase in the elastic property, such as tension or pressure, increases the wave speed, while an increase in the inertial property, such as linear density or volume density, decreases the wave speed.

Note what does not appear in the equations for wave speed: frequency and wavelength. Don't make the mistake of thinking that wave speed depends on frequency or wavelength. Confusion regarding this point arises because of the way textbooks traditionally write the formula, v = fλ. Mathematically speaking, it's conventional to put the dependent variable to the left of the equal sign. The formula v = fλ does not follow this convention. In order to follow the mathematical convention, the formula must be written as follows.

Equation	Proportionality	Statement
λ = v/f	λ α 1/f	The wavelength is inversely proportional to the frequency.

The statement given in the table doesn't mention the wave speed. That's because the wave speed is the constant of proportionality. In the relationship, λ = v/f, the wavelength is the dependent variable and is a function of the frequency, which is the independent variable. In many physical situations, we have control of the frequency. For example, when oscillating a string or a spring, we control the rate at which the medium is oscillating. When producing sound waves in air, we generate a particular frequency using a tuning fork, musical instrument, or other device. Note that it's the wavelength that depends on the frequency, not the wave speed. In none of these cases does the wave speed depend on the frequency. As mentioned previously, the wave speed depends on the elastic and inertial properties of the medium.

You may see the adjective non-dispersive used to describe a medium in which waves are propagating. A non-dispersive medium is one in which the speed of waves in the medium is independent of the frequency of the waves. This will be case for all the situations that we study in Chapter 14.