A string on a violin is pushed by a bow and the string vibrates at a frequency of 440 hertz. The sound waves produced by the vibrating string have a wavelength of 0.75 meters. What is the speed of the sound waves?
Okay, in this example, we have a violin and let’s just draw in one string on this violin. We’re told that a bow pushes across the string and the string starts to vibrate. The string vibrates at a frequency, we’re told, of 440 hertz. That means 440 cycles every second. So, if we were looking at a part of this vibrating string, one wavelength of the string, one instant it would look like this. And then, a half cycle of the wave later, it would look like this.
And we know the string will keep going back and forth, back and forth, between these two positions. And based on the frequency of the string, we know it makes one complete back and forth cycle 440 times every second. So, our string is vibrating at that frequency. And we’re told that the string has a wavelength of 0.75 meters. In other words, this distance here, from one end of the wavelength to the other, is three-quarters of a meter. Knowing that distance, perhaps the vibrating string on our violin would look more like this.
But anyway, we want to know what the speed of the sound waves on the string are. In other words, the wave that’s moving up and down the string and causing it to oscillate back and forth between these two opposite positions has some speed. To figure out what that speed is, we can recall a mathematical relationship between wave speed, wave frequency, and wavelength. If we call the speed of a wave 𝑣, then that speed is equal to the frequency of the wave, 𝑓, multiplied by its wavelength, 𝜆.
So then, the wave speed we want to solve for, and we can also call this 𝑣, is equal to 440 hertz, the frequency of our wave, multiplied by its wavelength, 0.75 meters. Before we multiply these numbers, let’s consider first what the unit of hertz means. Earlier, we said that the frequency of something is the number of cycles it goes through every second. And this means that another equivalent way to write the unit of hertz is to write it as one over seconds or inverse seconds.
When we write the unit as one over seconds, we can see what will happen with the final units of this calculation. In the numerator, we’ll have meters, and in the denominator, we’ll have seconds. That is, we’ll get meters per second, a speed in the units we want. 440 times 0.75 is 330. So, we find that the speed of the sound waves on this string is 330 meters per second.