# Wavelength

The distance from one peak to the next of a wave is called the wavelength. When the frequency of a simple harmonic motion is high, the wavelength is short. When the frequency is low, the wavelength is long.

Let’s take a look at this by watching the following video clip. In this video, the trace of a simple harmonic motion (hand movement) is drawn on the whiteboard with a pen. Let’s assume the wave propagation is simulated by moving the board at a constant speed. The trace of the pen is a sinusoidal wave. The wavelength is the distance between two adjacent peaks of the wave. Next, the frequency of the simple harmonic motion is faster. As a result, the wavelength is shorter. When the frequency is lower, then the wavelength is longer.

In the previous video, the speed of the moving board was always the same. This reflects that the speed of sound did not change. In the next video clip, the speed of sound will change, while the frequency of the simple harmonic motion is kept constant. We will see how the wavelength changes. First, the trace shows the original sinusoidal wave, when the board moves almost the same speed as in the previous video. Then, the speed of sound, that is, the speed of the moving board, gets faster. In this case, the trace shows us a longer wavelength. A shorter wavelength is observed when the board moves more slowly.

In sum, the wavelength is inversely proportional to the frequency of a simple harmonic motion, while the wavelength is directly proportional to the speed of sound. This is expressed by the following formula:

$$\lambda = c / f$$

where $\lambda$ denotes the wavelength (in m), $c$ denotes speed of sound (in m/s), and $f$ is the frequency.

The speed of sound in the air is approximated as 331.5 + 0.6 t [m/s], where t is temperature (in degrees centigrade). Therefore, we often use 340 m/s as the approximation of the speed of sound at room temperature. In this case, the wavelength of the pure tone of 1000 Hz is 34 cm.

1. Speaks, C. E., Introduction to Sound: Acoustics for the Hearing and Speech Sciences, Singular Publishing, San Diego, CA, 1999.