What is wave velocity?

Question

wavelength is distance travelled by wave in one time period of vibration when i first visualized this i thought it is total distance travelled by particles in one time period. but then i tried to understand this with a graph i assumed a particle to travel 1m from mean to extreme position in one second (completing its vibration in 4 seconds) and another particle which travels 1m in 2seconds (one vibration in 8 seconds)

with the help of graph i saw that with increase in time period the wavelength increases and when visualized the particles seems to slow down but still travel the same distance

so similarly how can i understand wave velocity with the help of graph ?

Answer 1

A wave is a signal passing through a medium. The signal is a pattern of oscillations of the elements of the medium. Each element oscillates back and forth, passing its portion of the signal on to the next element.

Frequency relates to the signal itself. It is the frequency at which each element of the medium oscillates. The amplitudes and relative timing of these oscillations (element by element) make the pattern.

Wave speed relates to the medium. It is based on the time required for each element to pass on its part of the signal to the next element.

Wavelength results from a combination of period (or 1/frequency) and wave speed. The period is the time required for an element to go through one cycle, the time required for one cycle of the wave to pass by a specific element of the medium. The wave speed multiplied by the time for a cycle to pass an element is the distance the wave travels during one cycle, the wavelength. This is why $\lambda=vT$, or $\lambda=\frac{v}{f}$. We often use the more convenient $f\lambda=v$, but $\lambda=\frac{v}{f}$ is more realistic. A medium that transports a signal more quickly (bigger wave speed) increases the wavelength. A higher frequency shortens the wavelength.

You can control frequency at the signal's source. You can control wave speed by controlling the medium. For sound, temperature and humidity matter. For a wave on a string, tension and mass density of the string matter. You cannot directly control wavelength. Sometimes you can filter signals so only certain wavelengths can be maintained, such as a standing sound wave within a trombone. Many different frequencies are generated. Only frequencies with wavelengths that match the instrument can build up to significant volume. All others almost immediately fizzle out.

Answer 2

Wavelength is not about the distance a particle has travelled. In the graph you provided, all particles do indeed travel the same distance because the amplitude of the wave/vibration is the same, thus the extreme values of the displacement are the same in all cases.

Wavelength is about the distance a wavefront travels until the position it started reaches the same phase again. In essence, this time interval is equal to the period $T$ of the wave.

To help the intuition a bit, just consider that you have "something" (this is the wave) that is travelling. From classical mechanics, the speed $c$ (I denote it $c$ because this is customary in wave theory) is equal to distance (or displacement) over the time interval. There you go, you have just found the speed equation for waves

$$c = \lambda f \tag{1} \label{1}$$

with $\lambda$ being the wavelength, which is equal to the distance travel in the time interval equal to one period $T$ and $f$ the frequency of the wave. You may argue that this doesn't look much like the classical equation for speed but if you change frequency to period due to $f = \frac{1}{T}$ into equation (\ref{1}) you'll get

$$ c = \frac{\lambda}{T} \tag{2} \label{2}$$

which looks a lot more like the $u = \frac{s}{t}$ equation.

Now, if you were to plot this you'd have to somehow track both distance and time. This would most probably result in a plot of amplitude versus distance with many snapshots (resulting in a moving image/animation) corresponding to various time instances. When the point from which the wave initiated completes a period then it is that you have to measure the travelled distance to find out what the wavelength is.

Answer 3

You cannot measure wave velocity by looking at the movement of one single particle in the medium. Technically, you cannot measure wavelength either, unless you already know or assume what the wave velocity is. You can measure the size of the particle's vibration, which determines the wave amplitude ($A$), and the time the particle takes to make one full cycle, which is the wave's frequency ($f$). But you cannot find the wave velocity or wavelength without more information.

Suppose you have a particle that, as you have said, "completes its vibration" in 4 seconds. Was that caused by a wave with a wavelength of 40 meters, moving forward at a velocity of 10 meters per second? Or a wave with a wavelength of 2 meters, moving forward at a velocity of 0.5 meters per second? Both would produce the same movement in that particle. There is no good way to distinguish between a long-but-fast wave and a short-but-slow wave.

This means that your existing graph only shows a change in wavelength if you assume that both waves have the same velocity. (Though this is actually a safe assumption in your case. More on that later.) The same graph could also represent two waves that have the same wavelength, but one wave is moving faster. Or some mix of the two.

In order to find the wave velocity ($v$), you need to measure the distance that the wavefront—the leading edge of the wave—travels over a given time period. You can then derive the wavelength ($\lambda$) from the velocity and the frequency.

$$\lambda = \frac{v}{f}$$

In practice, there are many cases where we already know what the wave velocity is because of prior measurements. Since you have tagged your question "acoustics", I will assume that we are talking about longitudinal waves, a.k.a. sound waves.

The velocity of a sound wave depends on the medium through which the wave is passing. Let's suppose it's air. If you know the temperature and composition of the air (including humidity), you can go and look up how fast sound waves will travel through that air. At 20 °C, it's about 343 meters per second. All sound waves will travel at that speed, even if they differ in amplitude (loud vs. soft sounds) or frequency (high vs. low sounds). There will be variation in wavelength, since that is inversely proportional to frequency, but not wave velocity.

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It is difficult to show any of this with a graph on paper. An animation would be better. Imagine an animated version of this:

...where the top wave was moving rather quickly, and the bottom wave was moving rather slowly. And imagine that there was a little dot on each wave, representing the movement of a single particle. You would see that the dots were moving up and down at the same rate. That would represent the tradeoff between wavelength and wave velocity.