The Wave Speed Equation

AnnaLet’s start with the basic question. What does the wave speed equation tell us?

MarcoIt connects three things: wave speed, frequency, and wavelength. In words, wave speed equals frequency multiplied by wavelength. Frequency tells us how many complete waves pass a point each second. Wavelength is the distance between matching points on neighbouring waves, like crest to crest.

AnnaSo if I know any two of those, I can work out the third?

MarcoExactly. The usual form is v equals f times lambda. Speed is usually in metres per second, frequency in hertz, and wavelength in metres. The units matter, so we need to make them consistent before calculating.

AnnaLet’s make that concrete. If the frequency is five hertz and the wavelength is two metres, what do we get?

MarcoMultiply five by two, and the speed is ten metres per second. If you wanted frequency instead, divide speed by wavelength. For wavelength, divide speed by frequency. One useful units check is that metres per second divided by hertz gives metres, since one hertz means one per second.

AnnaPeople sometimes hear “higher frequency” and assume that means a faster wave. Is that always right?

MarcoNo. For a wave travelling at a fixed speed in the same medium, frequency and wavelength move in opposite directions. Raise the frequency, and the wavelength gets shorter. Lower the frequency, and it gets longer. Their product still equals the speed. The speed depends on the medium and the type of wave, not on frequency alone.

AnnaAnd what happens if the wave crosses into a different medium?

MarcoIts frequency stays the same at the boundary, but its speed can change. The wavelength then changes to fit that new speed. That’s why we shouldn’t assume a wave keeps the same speed everywhere.

AnnaLet’s try the sound example. A sound wave in air has a frequency of four hundred forty hertz and a wavelength of zero point seven eight metres. How do we find its speed?

MarcoMultiply four hundred forty by zero point seven eight. That gives three hundred forty-three point two metres per second, or about three hundred forty-three metres per second to an appropriate number of significant figures. If we instead know the speed is three hundred forty-three metres per second and the frequency is five hundred hertz, dividing gives a wavelength of zero point six eight six metres.

AnnaWhat about light? The vacuum speed can look intimidating in scientific notation.

MarcoIt’s approximately three point zero zero times ten to the eighth metres per second. With a frequency of six point zero zero times ten to the fourteenth hertz, wavelength equals speed divided by frequency. The result is five point zero zero times ten to the minus seventh metres, which is five hundred nanometres. That’s in the visible range. But that vacuum speed applies only in a vacuum. Light travels more slowly in materials such as glass.

AnnaSo what’s the reliable routine when we face a problem?

MarcoWrite down what’s given, identify what you need, choose the equation or rearrange it, substitute, then calculate with units. Convert first if needed. For example, centimetres must become metres when you need speed in metres per second. And remember: multiply frequency and wavelength, don’t add them. If the answer needs a different unit, convert it before you finish.