What is the difference between standing wave and resonance?

[jinweiiii]

Hi, I am confused by the two concepts. How are they related? So, my interpretation is that a standing wave can happen without resonance. Resonance happens when a standing wave passes energy to another object, making it vibrate. Is that right? But some say a standing wave is an example of resonance?

 

[berkeman]

Welcome to PF.

Links to your reading please? Thanks.

 

[phinds]

What research have you done? What have you found out so far?

EDIT: I see berkeman beat me to it.

@jinweiiii this is not a Q&A forum. We are more interested in helping people figure out how to answer their own questions and we expect some effort from the person posting a question.

 

[berkeman]

What research have you done? What have you found out so far?

Great minds think alike...

 

[256bits]

But some say ...

Perhaps a reference expanding upon whom the 'some' might be,
My mind reading crystal ball purchased from the back alley street vender has once again failed to deliver.

 

[DaveE]

I think the short answer to your question is that standing waves depend on the boundary conditions. The length of a pipe (organ, flute, etc.) and whether the ends are open or closed determines the nature of the standing (acoustic) wave. This is sort of like constructive interference of reflected waves.

But a mass and spring have a resonant frequency that doesn't depend on initial conditions or boundary constraints. This structure has an innate resonance.

 

[tech99]

Hi, I am confused by the two concepts. How are they related? So, my interpretation is that a standing wave can happen without resonance. Resonance happens when a standing wave passes energy to another object, making it vibrate. Is that right? But some say a standing wave is an example of resonance?

Standing waves occur when two travelling waves of the same frequency pass each other in opposite directions. For instance, when sea waves strike a wall and are reflected back on themselves, or when radio waves are reflected by a metal plate. When we look at a standing wave it is not moving along but it is still going up and down, not frozen. Whereas a travelling wave is transporting energy as it goes along, a standing wave is not - rather it is a stationary store of energy.
The above examples do not exhibit resonance - they are not frequency sensitive in any way. But if we trap waves between two walls, then standing waves will occur only when the spacing allows the waves to exactly fit between them, and this is an example of resonance. Other examples of these transmission line resonances occur in organ pipes, waveguides and electrical cables.
Apart from a transmission line resonator, we can also have a lumped element resonator. For example, a mass on a spring, or a capacitor and inductor. In these lumped element oscillators we do not have standing waves.
If we have a standing wave resonator, such as a violin string, we can couple energy out of the standing wave into the body of the violin, which then acts as a sound radiator.
I hope this answers your questions.-

 

[nsaspook]

 

[Baluncore]

So, my interpretation is that a standing wave can happen without resonance. Resonance happens when a standing wave passes energy to another object, making it vibrate. Is that right? But some say a standing wave is an example of resonance?

That is all very confused.

Resonance involves circulating energy. For example, a pendulum resonates at its natural frequency, as energy is cyclically converted between potential and kinetic energy. If energy is gradually lost to the environment, the amplitude of the resonance will fall.

A standing wave requires a travelling waveform that repeats, and a reflector. The standing wave is formed, standing in place, by the sum of the forward travelling wave, and the reflected travelling wave.

A standing wave is not a resonance, but the repeating waveform that is reflected, may be generated by a resonator. There can also be more complex combinations.

Two reflectors, spaced to cyclically support a wave, travelling back and forth between them, can form a resonator. Standing waves may then be observed in the space between the two reflectors, where the two waves are travelling in opposite directions.

A resonator stores energy in a resonant cyclic process.
A standing wave is, the amplitude of, the sum of a wave and its reflection.

 

[Orthoceras]

A standing wave requires a travelling waveform that repeats, and a reflector. The standing wave is formed, standing in place, by the sum of the forward travelling wave, and the reflected travelling wave.

A standing wave is, the amplitude of, the sum of a wave and its reflection.

I would think the double slit interference pattern is a standing wave as well, because the nodal lines are stationary. In that case, reflection is not a requirement.

 

[Orthoceras]

Sorry, my mistake was already discussed and solved some time ago, here.

 

[A.T.]

But some say a standing wave is an example of resonance?

A standing wave can be amplified through resonance. But that doesn't mean that every standing wave is formed by resonance, or that every example of resonance involves standing waves (unless you stretch the meanings of the terms).

You can create a standing wave by initially deflecting the entire string in just the right way, and then letting it go. There is no further input sustaining the standing wave, so nothing to resonate to. The standing wave just dissipates over time.

You can create resonance with a single mass on a spring, by pushing it repeatedly at the right time. Here you don't have a wave at all, at least not one where you can tell the wavelength.

 

[Baluncore]

There is no further input sustaining the standing wave, so nothing to resonate to. The standing wave just dissipates over time.

A string resonates at its fundamental frequency, and at harmonics of that.
The frequency of the fundamental is determined by the string length, string tension, and the mass per unit length of the string. The string is a resonator, that was excited by your initial energy input.

You can create resonance with a single mass on a spring, by pushing it repeatedly at the right time. Here you don't have a wave at all, at least not one where you can tell the wavelength.

The height of the mass, plotted against time, is a wave. While it may not have a wavelength in space, it does have a natural frequency, and so a period in time.

A resonator does not need to be resonating, to be a resonator. It only needs to be able to cyclically store energy, for a number of cycles following excitation.

A standing wave does not store energy. The two waves, travelling in opposite directions, store and circulate the energy. The amplitude sum of those two waves, forms a pattern in space. That spatial pattern is called a standing wave.

 

[A.T.]

That's exactly what I meant by "stretch the meanings of the terms". If you stretch them far enough they become arbitrary and thus useless:

The height of the mass, plotted against time, is a wave.

Just because you can make a plot that looks wavy doesn't mean you have a physical wave.

A resonator does not need to be resonating, to be a resonator.

The question was about resonance (instances where something is actually resonating), not about something potentially being able to resonate.

 

[Baluncore]

That's exactly what I meant by "stretch the meanings of the terms". If you stretch them far enough they become arbitrary and thus useless:

The OP had already stretched the terms beyond breaking. There is no need to stretch the terms further while clarifying the terminology.

A standing wave can be amplified through resonance.

How is that possible?
Energy does not propagate in a standing wave. The amplitude of a standing wave is a direct function of the amplitude of the underlying travelling waves. Those two travelling waves do not require a resonator, just a single reflector.

 

[A.T.]

A standing wave can be amplified through resonance.

How is that possible?

You repeatedly push the string at the right time in the right direction.

 

[vanhees71]

Maybe it's time to leave the realm of wordy speculations and do some math ;-)).

The most simple example is the 1D wave equation, and so let's do the example of the string of length $L$ with fixed ends. The wave equation reads
$$(\partial_0^2-\partial_1^2) u(x^0,x^1)=0.$$
For simplicity I use relativistic notation, i.e., $x^0=c t$, and the string is along the $x^1$ axis; $\partial_{\mu}=\partial/\partial x^{\mu}$.

We look for the general solution of this equation with an arbitrary initial condition
$$u(0,x^1)=f(x^1), \quad \partial_0 u(0,x^1)=g(x^1)$$
with the boundary conditions
$$u(x^0,0)=u(x^0,L)=0.$$
To solve this we look for a complete set of eigenmodes, which we can use to expand our solution in terms of these eigenmodes.

To find these, we make the "separation ansatz"
$$u=A(x^0) B(x^1).$$
This gives
$$A''(x^0) B(x^1)-A(x^0) B''(x^1)=0 \; \Rightarrow \; A''(x^0)/A(x_0) =B''(x^1)/B(x^1).$$
Since the left-hand side is independent of $x^1$ and the right-hand side is independent of $x^0$ it must be a common constant, which we'll call $-k^2$:
$$A''(x^0)=-k^2 A(x^0), \quad B''(x^1)=-k^2 B(x^1). \qquad (*)$$
The general solution of the second equation is
$$B(x^1)=C_1 \cos(k x^1) + C_2 \sin(k x^1).$$
The boundary conditions now say
$$B(0)=0 \; \Rightarrow \; C_1=0$$
and
$$B(L)=C_2 \sin(k L)=0.$$
Since we cannot set $C_2=0$, because then we only get the trivial solution $u=0$, we must have
$$\sin(k L)=0 \; \Rightarrow \; k \in \frac{\pi}{L} \mathbb{N}.$$
The general solution for the 2nd equation then is most conveniently written as
$$A(x^0)= A_1 \cos(k_n x^0) + A_2 \sin(k_n x^0).$$
The general solution then is a superposition of these eigensolutions
$$u(x^0,x^1)=\sum_{n=1}^{\infty} \sin(k_n x^1) [A_{1n} \cos(k_n x^0) + A_{2n} \sin(k_n x^0)].$$
The $A_{1n}$ and $A_{2n}$ can be found from the initial conditions:
$$u(0,x^1)=f(x^1)=\sum_{n=1}^{\infty} A_{1n} \sin(k_n x^1).$$
Now the modes are a complete set of orthogonal functions in the interval $x^1 \in [0,L]$:
$$\int_0^L \mathrm{d} x^1 \sin(k_n x^1) \sin(k_m x^1)=\frac{L}{2} \delta_{mn}, \quad m,n \in \mathbb{N}.$$
Thus you get
$$A_{1n} = \frac{2}{L} \int_0^{L} \mathrm{d} x^1 f(x^1) \sin(k_n x).$$
The 2nd initial condition reads
$$\partial_0 u(0,x^1)=g(x^1)=\sum_{n=1}^{\infty} A_{2n} k_n \sin(k_n x^1),$$
leading to
$$A_{2n}=\frac{2}{L k_n}\int_0^{L} g(x^1) \sin(k_n x^1).$$
The general solution thus can be uniquely expanded in terms of the standing-wave modes
$$u_n(x^0,x^1)=\cos(k_n x^0) \sin(k_n x), \quad v_n(x^0,x^1)=\sin(k_n x^0) \sin(k_n x).$$

 

[DaveE]

Maybe it's time to leave the realm of wordy speculations and do some math ;-)).

The most simple example is the 1D wave equation, and so let's do the example of the string of length $L$ with fixed ends.

 

[phinds]

Well, the OP posted the question 4 days ago and then dropped out of the thread 20 minutes later and hasn't been back so it appears he's not really interested in the answer anyway.