Reflection of Waves and Formation of Standing Waves

In summary, the conversation discusses three doubts regarding waves on a string. The first doubt is about the inversion of a wave pulse when it reflects against a rigid boundary. The second doubt is about the equation of the reflected wave for both a rigid and non-rigid boundary. The third doubt is about the mathematical derivation of standing waves on a string fixed at both ends and fixed at one end only, and whether the same equations can be applied for longitudinal pressure waves. The conversation also includes a further explanation for the first doubt, which delves into the forces involved in wave propagation.
  • #1
FreezingFire
12
1
I have three doubts in regard to waves on a string which I will try to make as clear as possible.
For this purpose, I have considered a general wave:
$$y_i=y_0\sin(\omega t - kx)$$

(1) If a wave pulse:
$$y = y_0 e^{\frac{-1}{T^2} \left(t-\frac xv \right)^2}$$
is incident against a rigid boundary, such as a fixed wall to which the string is attached, why does it get inverted when it is reflected? I know that as the wave hits the boundary, considering the wave pulse traveling from left to right, the left part pulls the element of string fixed to the wall upwards, and the wall exerts an equal and opposite force on it, thus preventing the element from moving. Due to this force, the left part of string is pulled downwards upto its mean position, beyond which it (according to me) moves downwards due to inertia. My question is, at this stage, won't the left part of the string pull the said element downward again? Wouldn't the wall again generate a new pulse in the upward direction? Since this doesn't really happen, what is the correct reasoning behind the inversion of the pulse? Also if possible, could similar reasoning be applied for reflection against a non-rigid (soft) boundary? (View attached image)

(2) Now considering the initial sine wave, if it reflects against the rigid boundary, what will be the equation of the reflected wave? As far as I know, it must be:
$$y_r=y_0\sin(\omega t + kx + \pi)$$
or,
$$y_r=-y_0\sin(\omega t + kx)$$
Is this correct? And what about reflection against a non-rigid (soft) boundary? Is it the following?
$$y_r=y_0\sin(\omega t + kx)$$

Also, how do we obtain these equations (short proofs or conceptual proof)?

(3) In our textbook, the standing waves were explained as superposition of two waves traveling in the opposite direction, where one was the incident wave (given as ##y_i=y_0\sin(kx - \omega t)## in the book), and the other was reflected from either a hard boundary or a soft one. But in both cases, the equation of standing wave used is exactly the same, i.e.:
$$y=2y_0\sin(kx)\cos(\omega t)$$
where component waves were given as ##y_i=y_0\sin(kx - \omega t)## and ##y_r=y_0\sin(kx + \omega t)##. This meant that my equation for reflection was wrong too, as ##y_r## is different in this case! Also for soft boundary the exact same equation of standing wave was used, meaning the reflected wave was also the same in both cases! Clearly this isn't possible. So what is the actual mathematical derivation of standing waves on string fixed at both ends, and string fixed at one end only?

Finally, can we apply all these results for a longitudinal pressure waves (i.e. sound)?

Thanks in advance!
 

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  • #2
Maybe I can explain it this way.

You have a cord where a pulse is generated at one end "A" and then travels to the other end "B" where the cord is solid fixed to a wall.

But let's look at a similar situation: You have a cord where a pulse is generated at one end "A" and then travels to the midpoint "B" where it meets another inverted pulse traveling from the opposite direction an originating at "C", the other endpoint. In such a case, the pulse traveling from A to C will pass right through the pulse traveling from C to A - each escaping unchanged once the pass the center point B and each other.

But what would they look like at B? How would point "B" move? The answer is, it wouldn't. The two waves cancel each other out at that point. It is exactly as if "B" was fixed to a stationary wall.
 
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  • #3
Thanks for the detailed answer, but I already knew this explanation from our textbook! What I further need is how do we explain this in terms of forces only [this is for doubt (1)]. I now realize that maybe this delves deep into how a wave actually propagates on a string in terms of forces...!
 
Last edited:
  • #4
Anyone there? Please help me with all my three doubts!
 
  • #5
Also, I see that perhaps my doubt #3 is a bit confusing. So I will reword it:

1. Standing waves on string fixed at both ends
Consider a wave ##y_i = y_0 \sin(kx - \omega t)## on a string fixed at ##x=0## and ##x=L##.

As the incident wave gets reflected from the fixed end at ##x=L##, the equation of reflected wave is ##y_r = y_0 \sin(kx + \omega t)##, as the reflected wave in this case travels in the negative x direction and has a phase difference of ##\pi## with the original wave. These two waves interfere to form the standing wave:
$$y = 2y_0 \sin(kx) \cos(\omega t)$$
This is correct as far as my textbook is concerned.

2. Standing waves on string fixed at one end
Again consider the same wave ##y_i = y_0 \sin(kx - \omega t)## on a string fixed at ##x=0## and attached to a light string (acting as a free end) at ##x=L##.

Now, the incident wave reflects at ##x=L## as ##y_r = -y_0 \sin(kx + \omega t)##, as it has no phase difference with respect to the original wave, and it travels in the negative x-direction. These two waves would interfere to give the standing wave:
$$y = -2y_0 \cos(kx) \sin(\omega t)$$
This would imply that ##x=0## is an antinode, which is clearly impossible. So where am i going wrong? I think the error might be in the italicized sentence. Also, our textbook has directly given us the equation of standing wave in this case also as:
$$y = 2y_0 \sin(kx) \cos(\omega t)$$
which is same as for a string fixed at both ends.
This is why i was actually asking for the equations and derivations of reflected waves in doubt #2.
 
  • #6
Anyone there?
 
  • #7
Can someone please help me with these questions?
 

Related to Reflection of Waves and Formation of Standing Waves

1. What is reflection of waves?

Reflection of waves is the phenomenon where a wave bounces off of a surface and changes direction. This occurs when a wave encounters a boundary or obstacle that does not allow the wave to pass through.

2. How is the reflection of waves related to the formation of standing waves?

The reflection of waves is essential in the formation of standing waves. When a wave reflects off of a boundary, it combines with its incoming wave and creates a standing wave pattern. The standing wave is formed when the reflected wave and the incoming wave have the same frequency and amplitude.

3. What factors affect the formation of standing waves?

The formation of standing waves is affected by the frequency, wavelength, and amplitude of the wave, as well as the properties of the medium it is traveling through. The length and shape of the boundary or obstacle also play a role in determining the standing wave pattern.

4. Can standing waves only form in one-dimensional systems?

No, standing waves can form in both one-dimensional (e.g. a string) and two-dimensional (e.g. a membrane) systems. However, the standing wave patterns will differ depending on the dimensionality of the system.

5. How are standing waves used in real-world applications?

Standing waves have various practical applications, such as in musical instruments, where they create resonance and produce specific musical notes. They are also used in medical imaging techniques, such as ultrasound, and in optical devices, such as lasers.

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