How Do Waves Propagate on a String?

[Greg Bernhardt]

Definition/Summary

Take the case of a string with it's one end [which will refer to as the 'fixed end'] tied to a wall. The other end [which we will refer to as the 'free end'] is held by you. It is a common experience that if we move that free end, a 'bump' travels along the string. And if we continuously move the free end up and down, a 'train of pulses' travels along the string. It is also a common experience that a small part of that 'bump' comes traveling back, as if it were 'reflected' by the wall. This article deals with this phenomena and it's far reaching applications.

A Wave, as is defined, is a disturbance that propogates through space and time. In the case, of waves on a string, the 'propogation' occurs through a medium i.e. the String and the disturbance that we talk about is the upward displacement of each particle.

This article also discusses some points which are general to all waves. Hence, this article is something like an introduction to waves.

Equations

The following set of equations are for reference purposes. Please see extended explanation for notes on the following equations.

i] Velocity of a wave on a string:

$$v = \sqrt{\frac{T}{\mu}}$$

where, $T$ is the Tension in the string and $\mu$ is the linear mass density of the string.

ii] Average power transmitted across any point:

$$P_{av} = 2\pi^2f A^2 v \mu$$

where, $A$ is the Amplitude, $\mu$ is the linear mass density of the string, $v$ is the speed of the wave and $f$ the frequency.

Extended explanation

1. Informal approach to Waves on a String

What happens when you move the free end of the string is that, you cause an 'upward displacement' of the immediate particle at the free end. This particle while traveling upwards, pulls the particle next to it upwards. There is a little delay in this pull and hence, all the next particle's upward displacement, at that point of time however, is a little less than for the first particle. This particle of the string now pulls the particle next to it and so on.

When the first particle reaches the topmost height [also known as it's amplitude], and then you pull it down, this particle pulls down the particle next to it. Again, owing to the same delay, the next particle, at this point of time is continuing it's upward motion, and hence reaches the same topmost height before it is pulled down. This particle does the same thing to the next particle and so on, the 'bump' travels.

http://img364.imageshack.us/img364/5400/wave1ys4.jpg

During this wave motion, it is seen that the particles of the string never leave their position on the string i.e. if a particle is at a length 'l' of the string from the fixed end, it will always be at length 'l' of the string from the fixed end. If the fixed end were fixed to some energy converter which converted mechanical energy to other forms of energy, we can see that we have 'transferred' energy from the free end to the fixed end i.e. across a distance, but without no displacement of a particle. No 'matter' had to travel to transfer this energy. The string acted as a medium across which mechanical energy could be transferred, just as Electric lines transfer electrical energy [this is also a wave phenomena, but in this article is treated like an analogy]. This was a surprising result to the physicists as the transfer of energy without the transfer of any velocity bearing object was unheard of.

2. Equation of a traveling wave

Let's say that you starting moving the free end at time t=0. And you finish doing so, at time $t = \Delta t$. The vertical displacement of the particle, at t=0 is y = 0, and during the time between 0 and $\Delta t$ is non-zero. Also, any time after $\Delta t$, the displacement is y = 0. Clearly, this function is a displacement of time. Let's state this mathematically as:

$$y = f(t)$$

The function f(t) represents the displacement 'y' of the particle just at the free end, as a function of time. Let us now assign a coordinate system to this setup: The free end is the origin, and the length of the string, when no oscillations take place as the positive x-axis. Then,

$$y(x=0, t) = f(t)$$

Let us say that this 'bump' travels along the string with a velocity 'v'. If a certain displacement of the free end occurred at a time 't', then it will reach a point 'x' on the string at a time (t + x/v). Which means, that if a displacement occurred at a time 't' at a point 'x' on the string, it occurred at the free end at a time (t - x/v). What this means is that, a point 'x' seems to 'mimic' the displacements of the free end with a time delay of 'x/v'. What happens at the free end at some time, happens at a point 'x' time 'x/v' later. Hence, the displacement of the particle at 'x' is the same as the displacement of the particle at the free end at time 't - x/v'. From out previous definition, this displacement will be given by:

$$y = f\left(t - \frac{x}{v}\right)$$

This is the equation of a traveling wave. The function 'f' is arbitrary, which should be obvious as one is free to chose how he moves the free end.

A thing to note here is that, if the velocity is very high, then f(t - x/v) approximates f(t). What this means is that, a particle at 'x' mimics the particle at the free end almost instantaneously. This is the basic definition of wave velocity i.e. the higher the velocity, the quicker is a particle at 'x' displaced in the same manner as the particle at free end. You must have noticed, that in a string which is held tightly, the particles in the string move up quickly as the first particle is moved. However, in a slack string, the particles away from the free end take a longer time to show this displacement. Wave velocity on a string is indeed dependent on the tension in the string, as we shall soon see.

3.Sine wave on a string

A very important case of Waves on a string is when the free end is moved in Simple Harmonic Motion i.e.

$$y(x = 0, t) = A \sin(\omega t)$$

This wave propagates along the string, such that each particle performs Simple Harmonic Motion, with a time delay w.r.t to the particle at free end, 'x/v'. Here, 'v' again, is the wave velocity. Hence, the wave equation is given as:

$$y = A \sin\left(\omega t - \frac{x}{v}\right)$$

This follows from how we defined a wave equation in Section 2. Note here, that this function, gives the displacement of any particle, at a time 't' and position 'x'. For example, if there is a particle at x = 1m, then the displacement of the particle as a function if time is given by:

$$y = A \sin \left(\omega t - \frac{1}{v}\right)$$

Here, the term $-1/v$ is known as the phase difference for the particles of the string in SHM. The phase difference is what we talk about when we talk about the 'delay' in mimicking the displacement of the particle at the free end. Clearly, this particle performs SHM too, so does every other particle. Do note that the value of the Sin function lie between -1 and +1. Hence, the displacement caused of each particle in this wave will lie between -A and +A.

Now, let's say, we would like to see how the wave looks at t = 5s. It is simply given by:

$$y = A \sin \left(5\omega - \frac{x}{v}\right)$$

This is the 'shape' of the string at a particular time. As is obvious, this equation gives us the relation between the x-coordinate and y-coordinate of a path. At different times it differs, but when we talk about a particular instant, substituting 't' for that particular instant gives is the shape of the wave.

http://img364.imageshack.us/img364/9666/wave2iy5.jpg

As you can see, the wave is a real-life impression of a plot of the Sine function!

We know that the particles in the string travel only and only in the direction of the y-axis. Hence, if we are to compute the velocity of a particle at a point 'x', we need to differentiate 'y' w.r.t time and we shall get it's true velocity i.e. since there are no other velocity components to this particle, we can treat 'x' as constant while differentiating 'y' w.r.t time. We do this because, for a particle at a point 'x', during a short time $dt$, 'y' changes, but there is no change in 'x'. Doing so gives us:

$$\frac{\partial y}{\partial t} = A \omega cos \omega (t - x/v)$$

The direction of the velocity of the particle is not 'in phase' as the particle. We shall come back to this topic when we discuss about Angular parameters of a Sine Wave.

4. Terms associated with a Wave

i. Amplitude

We know that a sine wave on a string is given by the equation:

$$y(x, t) = A \sin (\omega t - x/v)$$

Here, 'A' is the Amplitude of the wave i.e. the maximum displacement of any particle. This is same as the Amplitude of the SHM that the source at the free end performs.


ii. Time period and Frequency

Simplifying the wave equation, we get:

$$y(x, t) = A \sin(\omega t - kx)$$

where, $k = \omega / v$.

Here, 'omega' is known as the 'Angular frequency' of the wave and 'k' is known as the 'Angular Wavenumber'. We will soon see what these terms mean.

Now, an important property of the Sine Wave is that the motion is 'periodic' i.e. the motion repeats itself after a certain time interval 'T'. Since, every particle executes SHM, we know that this is true. What this means if that if a particle at 'x', is displaced by 'y' at a time 't', then it will have the same displacement at a time 't + T', 't + 2T' and so on. This time period 'T' is known as the 'Time Period' of the oscillation. Now, we will find out what the quantity 'T' is. Let us take a fixed point 'x'. It's displacement at time 't' is given by:

$$y = A \sin(\omega t - kx)$$

At a time (t + T), the displacement is given by:

$$y = A \sin(\omega (t + T) - kx)$$

On equating these values, we get:

$$\omega t - kx + 2n\pi = \omega (t + T) - kx$$

(This comes from the periodicity of the Sine function). Hence,

$$T = n\frac{2\pi}{\omega}$$

Since, the 'Time period' is given as the smallest time interval in which the motion repeats itself, we put n = 1 in the above equation to get:

$$T = \frac{2\pi}{\omega}$$

This is the 'Time period' of the wave. Another parameter 'frequency' is defined as:

$$f = \frac{1}{T} = \frac{\omega}{2\pi}$$

While the Time period suggests the time taken for a single cycle, Frequency tells us as to how many cycles are completed per unit time.

NOTE: A very important note one should make here is that, as we saw the frequency of the wave is same as the frequency of SHM of the Source. The source hence, is free to choose any frequency of oscillation regardless of what the string is made up of, how long it is or any other parameter associated with it. Hence, for a wave, the frequency remains the same regardless of the medium. However, the speed and the wavelength do not. They change in such a manner so that the frequency remains constant. A relation between frequency, wavelength and velocity is given in the next section.

iii. Wavelength and Wavenumber

Now, let us consider the shape of the wave at a particular instant, $\tau$:

$$y = A \sin(\omega \tau - kx)$$

Take any point 'X' on the string wave. There are many other points which have the same displacement as this particular point. The distance from 'X' of the nearest such point is known as the Wavelength, $\lambda$ of the wave. All the points having the same displacement as 'X' are at a distance $n\lambda$ from X, where 'n' is an integer. Using the same math as we did in the previous section, we get:

$$\lambda = \frac{2\pi}{k}$$

If we break the wave into small parts, such that each part is wavelength long, you will see that, if we take any part and repeat it on either side, we can construct the full wave. This small part is a 'pulse'. Also, the first particle and the last particle of this pulse always have the same displacement. Such particles which always have the same displacement, irrespective of time are said to be 'in phase'. What 'phase' means shall be dealt with in the next section.

http://img175.imageshack.us/img175/1029/wave5wb9.jpg

Here, 'k' is known as the cyclic wavenumber. Another parameter that we define here is 'Wavenumber'. Wavelength tells us about how long one pulse of a wave is. Wavenumber, on the other hand is the measure of no. of pulses per unit time:

$$\bar \lambda = \frac{1}{\lambda} = \frac{k}{2\pi}$$

iv. Frequency and Wavelength

Let's say that a wave is traveling with a velocity 'v'. Hence, one cycle of the wave is completed in 1 time period i.e. T. Also, the length covered during this time is the Wavelength since, the Wavelength equals the length traveled by a wave in 1 cycle. Hence, we have:

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

5. Angular Parameters of a wave

One of the difficulties faced by students is understanding the physical significance of the term 'angular' in 'angular frequency'.

Consider a position 'x' on the string. As we know it is performing SHM, with it's displacement given by:

$$y = A \sin(\omega t - kx)$$

Let us take a point which moves in a circle, whose radius is equivalent to the Amplitude of motion and it's center is 'x' [which we shall treat as the origin for now]. The y-coordinate of this point is given as: $A \sin(\theta)$, where $\theta$ is the angle the position vector of the point makes with the positive x-axis. If it were to rotate such that it's projection on the y-axis gives the position of the particle at 'x', we will find that, the angular velocity of this point is nothing but the Angular Frequency of the SHM. And hence the word 'Angular'.

http://img175.imageshack.us/img175/8848/wave6wc4.jpg

In the image above, the green bob is the particle of the string that oscillates at the position 'x', which in this case we treat as the origin.

Phase

An important parameter with waves is the 'phase'. Let there be two particles performing SHM on a string. They are separated by a distance 'l'. Their displacements at a time $\tau$ are given as:

$$y_1 = A \sin(\omega \tau - kx)$$

$$y_2 = A \sin(\omega \tau - k(x + l) = A \sin(\omega \tau - kx - kl)$$

Here, both the equations are identical, other than for the fact that there is an extra term, 'kl' for the second particle. This term is known as the Phase difference for the two particles. Phase is a relative term. It is only significant when we talk about two or more oscillations and/or waves. Saying that Phase of a wave is $1$ or $2$ is meaningless. Phase is an angle and we generally deal with phase differences which are factors of $\pi$. Coming back to our previous case of a rotating point.. assume that there are two such circles at each of the point separated by a distance 'l'. Then, the angle made by the two position vectors at any time gives us the 'phase difference' of the two oscillations.

The difference in the displacements of the two particles at any instance is known as their 'path difference'.

If the length 'l' equals the Wavelength of the wave i.e. $l = 2\pi / k$, we have:

$$y_2 = A \sin(\omega \tau - kx - 2\pi) = A \sin(\omega \tau - kx)$$

which is the same as $y_1$. This verifies our statement that particles at a separation of an integral multiple of Wavelength are always in tandem i.e. they are 'in phase'.

Suppose, a wave has the equation:

$$y = A \sin(\omega t - kx)$$

Another wave has the equation:

$$y = A \cos(\omega t - kx)$$

For the second wave, we know that $\sin(x + \pi / 2) = \cos(x)$. Hence the second wave is know given as:

$$y = A \sin(\omega t - kx + \pi / 2)$$

Hence, now the 'phase difference' of the two waves is $\pi / 2$.

http://img364.imageshack.us/img364/9707/wave3zk4.jpg

We now arrive at the general equation of a Sine Wave:

$$y = A \sin(\omega t - kx + \phi)$$

Here, $\phi$ is the phase of the wave. Supposing, we take the time t = 0 to be when the source completes exactly half an oscillation. Which means that a pulse of half the wavelength has already travelled. Which means that, the SHM of the free end will differ with the rightmost particle of the pulse by a phase of $\pi$. Putting that into the equation, we get the Sine Wave equation as:

$$y = A \sin(kx - \omega t)$$

6. Interference of Waves

Let us assume that even the fixed end of the string now is not fixed. It is free and some other person moves it in SHM. The displacement caused by the first person is given as:

$$y_1 = A_1 \sin(k x - \omega t)$$

And the displacement caused by the second person is given as:

$$y_2 = A_2 \sin(k x - \omega t + \phi)$$

Since the phase is relative, we take the phase of the first wave to be 'zero'. Hence, \phi automatically becomes the phase difference of the two waves. The net displacement of any particle 'x', at a time 't' is given as:

$$y = y_1 + y_2 = A_1 \sin(k x - \omega t) + A_2 \sin(k x - \omega t + \phi)$$

This is known as the 'principle of superposition'. The resultant is also a Sine Wave. This result can be derived using elementary Algebra. The resultant wave is given as:

$$y = A[\sin(kx - \omega t + \epsilon)]$$

where,

$$A = \sqrt{A_1{}^2 + A_2{}^2 + 2A_1 A_2 cos(\phi)}$$

and,

$$tan(\epsilon) = \frac{A_2 \sin(\phi)}{A_1 + A_2 \cos(\phi)}$$

The resultant wave is a sine wave only and only if the Frequency and Wavelength of the wave is same for all the sources. This resultant wave seems very similar to vectorial addition.

So, as a simplifying tool, each wave is represented as a Vector. Where, the length of the vector is given as the amplitude of the wave and it's angle is it's phase. As such, the resultant wave of two such waves is the resultant vector of two such vectors.

http://img175.imageshack.us/img175/5486/wave4bj0.jpg

Another point to note here is that the resultant amplitude lies between, $A_1 + A_2$ (maximum) and $|A_1 - A_2|$ (minimum). Both these cases correspond to the phase difference being $\phi = 2n\pi$ (max amplitude) and $\phi = (2n + 1)\pi$ (minimum amplitude).

The first case is said to be the case of 'Constructive Interference'. And the second case is said to be the case of 'Destructive Interference'.

7. Refelection of Waves

Assuming that the end of the wave is fixed, when a wave pulse travels and hits the fixed end, a reflected wave is generated, whose first pulse is inverted as compared to the last pulse hitting the wall.

This happens because the last oscillating particle, exerts a force on the wall in, let's say a downward direction. The wall then, exerts a force on that particle in the upward direction [as they are Third Law pairs]. This force causes the wave pulse to travel in the backward direction again in the form of a wave. This phenomena is known as 'Reflection of Waves'.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!