# How to correlate the frequency of source to The frequency of particle just between the two medium while transmission and reflection of wave?

When a pulse travels through rarer to denser medium,two new waves (reflected and transmitted) are formed from incident wave,While the wavelength of the Transmitted wave differs from wavelength of Incident wave,also velocity is decreased,But the frequency of incident,Reflected and transmitted ray remains same

I cannot figure out why does this happen, There are many answers on physics stack exchange forum whose essense is as follows(or they explain it as follows)

1.(Why doesn't the frequency of light change during refraction?) An answer here(for light wave) states that wave has to follow boundry conditions, i.e to remain in phase to match electric fields

2.similiarly Why does frequency remain the same when waves travel from one medium to another? An answer here states example of swimming pool,Which itself Referrs to boundry condition that particle just above and just below water surface must oscillate with same frequency so that rope would not break

Similarly,Other answers at such question indirectly refer to a boundry condition to be satisfied

My question is that,during the transition process of incident wave to reflected and transmitted wave,we can call the Particle at common interface of two medium of rope as source of these new reflected and Transmitted wave so we can deduce that reflected and transmitted wave must have a common frequency,

But why should this common frequency must equal the Frequency of incident wave??As you can see in the animation attached below,During transition,In first medium,The final wave formed(during transition) is superposition of Incident and reflected wave,Final wave has to follow the frequency of Interface particle(we called it new source earlier)  On a note,I understand that superpositioned wave in first medium,Reflected wave and transmitted wave must have common frequency,But I can't understand why this common frequency must equal frequency of incident wave

This is the link of animation stated in question

https://surendranath.org/GPA/Waves/TWRT/TWRT.html (please select thinner to thicker option in website above)

Because you have assumed a linear time invariant medium. A linear time invariant medium is described by a linear time-invariant operator whose eigenfunctions are pure sinusoids that are characterized by their amplitude, frequency and phase. Only amplitude and phase may be changed by such system, meanwhile frequency stays the same between input and output. This is true for lumped as well as for distributed systems, such as RLC circuits or dielectric crystals, resp. This is not really physics. The physics question is rather why a material medium through which an EM wave propagates, for example glass or crystalline dielectric, is linear and time invariant? In practice, nothing is truly linear nor time invariant. For large enough EM intensities in the nonlinear regime you get as minimum harmonics and may also unplanned for nonlinear modulation on the incoming wave and thus shift or spread its fundamental frequency.

If the medium is nonlinear and/or time varying the frequency of the transmitted wave will change. For example, a linear time varying medium is a crystal in which acoustic waves (sound) modulate the density and thus the dielectric permittivity of the medium and when light is scattered off the acoustic wave the the scattered light suffers frequency shift that can be used to measure the the acoustic frequency (an instantaneous spectrum analyzer, see 1.

As hyportnex mentioned, it is a linear system and in that case the frequency won’t change. If it a nonlinear material you can have all kinds of things happen like frequency doubling or tripling.

Or if you look closely at other situations you can have other types of scattering like Raman or Brillioun scattering where an interaction with the lattice vibrations (phonons) in material can result in a change in frequency and direction.

So a more general way to think about your question is that energy and momentum need to be conserved. This is always true. With photons you have to be a little careful since they are massless and in the special cases like Raman scattering it is essentially keeping track of the energy of the photon the energy that is lost to the molecule vibration and the momentum conservation is taken care of by change and of direction and phonon vibration.

But for your case you can also calculate the energy of your incident wave and the momentum of that wave and the energy and momentum of the transmitted and reflected waves. And the Total Energy would be equal to the the sun of the reflected energy and transmitted energy.

If there was a change in frequency of one of the waves then energy would not be conserved.

If you come in at an angle to the surface of some refractive index then you then match up the vector components to determine the angle of refraction, and the angle of reflection being equal to the angle of incidence. This is essentially ensuring the momentum is conserved.

In the medium with the refractive index, the wavelength does change, but the frequency remains constant. The medium doesn’t add or subtract any energy to the wave since in the linear system and energy is conserved. The change in wavelength is also important in keeping the phase correct.

When you start dealing with nonlinear systems, you still have to have conservation of energy and momentum…but it is more complicated to do the math.

Just think of yourself as a particle at the boundary.
An incident wave of frequency $$f$$ forces a nearest neighbour and hence you to oscillate at the same frequency $$f$$ .
You are connected to nearest neighbours via bonds.
How can you force those nearest neighbours to oscillate at any other frequency other than frequency $$f$$?
The answer is that you cannot, and so as a result of your oscillations at frequency $$f$$ your nearest neighbours must also oscillate at a frequency $$f$$.

• what if there is just one pulse of complete sinuside(up+down),and I say (Time)^-1 for one particle of that rope to reach its original position after passage of wave be its frequecy,and not continuous source to produce constant frequency,why can't now the incident wave frequency change after reflecting back,or interacting with first particle of other medium due to reaction force? Sep 12 at 7:51
• You have now changed the question to having a pulse of a certain duration arriving at the interface and then another pulse arriving with a different duration. There is nothing wrong with that and the particle at the boundary will do exactly what it is told to do by the incoming pulses. Sep 12 at 8:28
• if I may rephrase my doubt,once again,I would really like to know that suppose we need to change timeperiod of pulse,we can do so by applying force to one of its end particles,similiarly,when no interaction is done to pulse travelling with same initial time period and suddenly it faces a dense medium particles,it will have reaction force acted,which will make two pulses one transmitted forwad and one reflected,why then then even after reaction force acted by dense particle on rarer particle still don't change time period?Is there some way to see this using newton mechanical analysis? Sep 12 at 8:34
• The dense and rarer particles are linked with a bond. If they start having different periods that bond will have to break. Sep 12 at 8:44
• I am not implying that they should have different periods,I know that they must have same frequency in order for prevention of bond breakage,but I am asking that before interaction,rarer particle had some time period let's say $t_1$,denser particle had time period $0$,now after interaction ,let's say their mutual time period is $t_2$,all I am asking is that why is it necessary that $t_2$ must equal $t_1$ Sep 12 at 9:02

It follows from the algebraic conditions governing interface condition at the interface between the two media, with linear behavior.

## Details

You can think at the problem as the union of the problems in two neighboring media and a common boundary. Let's call medium 1, the medium where the original incident wave is traveling and reflection occurs, and medium 2 where the transmission occurs. Medium 1 is in the region of space $$M_1: x \in [x_1, x_i]$$, medium 2 in $$M_2: x \in [x_i, x_2]$$, where $$x_i$$ is the position of the interface.

Interface conditions. What happens in the media depends on the common boundary surface, and what happens on both sides of the boundary is governed by interface conditions (coming from integral balance equations, for an elementary volume across the interface surface). As an example, the electric field on a interface with no charge surface density is constant across the surface

$$\mathbf{e}_1(x_i,t) = \mathbf{e}_2 (x_i,t)$$.

At the medium interface, usually an algebraic condition holds to determine the transmission and reflection ratio, namely

$$i_t = 1 + i_r,\qquad$$ with $$i_t \in (0,1)$$,

that quantifies the amount of the signal transmitted in medium 2 and the amount of signal reflected, as fraction of the incident signal.

Solution as superposition of traveling waves. The field in medium 1 is the superposition of the incident and the reflected wave, while field in medium 2 come from the transmitter wave only

$$e_1(x, t) = e_i(x,t) + e_r(x,t) = e^{t.w.}_i(x-v_1t) + e^{t.w.}_r(x+v_1t)$$
$$e_2(x, t) = e_t(x,t) \qquad \qquad \ = e^{t.w.}_t(x-v_2 t)$$,

having written them explicitly as traveling waves. Now we can write the three contributions as:

• incident wave: its value at the interface is $$e^{t.w.}_i(x_i -v_1 t)$$.
• transmitted wave: its at the interface is $$e^{t.w.}_t(x_i - v_2 t) = i_t e^{t.w.}_i(x_i - v_1t)$$,
• reflected wave: its at the interface is $$e^{t.w.}_r(x_i - v_2 t) = i_r e^{t.w.}_i(x_i - v_1t) = (i_t - 1) e^{t.w.}_i(x_i - v_1t)$$,

so that both the interface condition $$e_1(x_i, t) = e_2(x_i, t)$$ and the ratio of the transmission and reflection signal are satisfied.

Properties of traveling waves. Now, we only need to reconstruct the full signal in the media, knowing that its made of traveling waves and knowing the boundary conditions. For a "backward" traveling wave, we know that

$$f(x_0,t_0) = f^{t.w.}(x_0+vt_0) = f^{t.w.}(x_1+vt_1) = f(x_1,t_1)$$,

if the argument of the traveling wave is the same, i.e. $$x_0 + v t_0 = x_1 + v t_1\qquad \rightarrow t_1 = t_0 + \dfrac{ x_0 - x_1}{v}$$,

while for a "forward" traveling wave $$t_1 = t_0 - \dfrac{ x_0 - x_1}{v}$$.

Solution of the problem. Putting everything together, we can write the solution of our problem. The field in medium 1 reads

$$e_1(x,t) = e_i(x,t) + e_r(x,t) = \\ \qquad \ \ \ = e^{t.w.}_i(x-v_1t) + e^{t.w.}_r(x+v_1t) = \\ \qquad \ \ \ = e^{t.w.}_i(x-v_1t) + e^{t.w.}_r\left(x_i, t - \dfrac{x_i-x}{v_1}\right) = \\ \qquad \ \ \ = e^{t.w.}_i(x-v_1t) - (1-i_t) e^{t.w.}_i\left(x_i, t - \dfrac{x_i-x}{v_1}\right) = \\ \qquad \ \ \ = e^{t.w.}_i(x-v_1t) - (1-i_t) e^{t.w.}_i\left(x, t - 2\dfrac{x_i-x}{v_1}\right) \\ \qquad \ \ \ = e^{t.w.}_i(x-v_1t) - (1-i_t) e^{t.w.}_i\left(x -v_1 \left( t - 2\dfrac{x_i-x}{v_1} \right) \right) \\ \qquad \ \ \ = e^{t.w.}_i(x-v_1t) - (1-i_t) e^{t.w.}_i\left(-x -v_1 \left( t - 2\dfrac{x_i}{v_1} \right) \right) \\$$

while the field in medium 2 reads

$$e_2(x,t) = e_t(x,t) = \\ \qquad \ \ \ = e^{t.w.}_t(x-v_2t) = \\ \qquad \ \ \ = e^{t.w.}_t\left(x_i, t - \dfrac{x-x_i}{v_2}\right) = \\ \qquad \ \ \ = i_t e^{t.w.}_i\left(x_i, t - \dfrac{x-x_i}{v_2}\right) .$$

Harmonic signals. Assuming the incident signal is an harmonic signal, $$e^{t.w.}_i(x-v_1 t) = A e^{i k_1(x-v_1t)} = A e^{i (k_1x-\omega_1 t)}$$, with $$\omega_1 = \dfrac{v_1} k_1$$,

$$e_1(x,t) = A e^{i (k_1x-\omega_1 t)} - (1-i_t) A e^{i \left(-k_1 x -\omega_1 \left(t -2\frac{x_i}{v_1} \right) \right)} = e^ {i \omega t} \left[ A e^{i k_1 x} - (1-i_t) A e^{i \left(-k_1 x + 2 \omega_1 \frac{x_i}{v_1} \right) } \right]$$

$$e_2(x,t) = i_t A e^{i \left(k_1 x_i -\omega_1 \left( t - \frac{x-x_i}{v_2} \right) \right) } = e^{-i \omega_1 t} \left[ i_t A e^{i \left(k_1 x_i + \omega_1 \frac{x-x_i}{v_2}\right)} \right]$$,

that clearly shows that, at every point in space, the signal has the same time frequency $$f_1 = \dfrac{\omega_1}{2\pi}$$.

You say :superpositioned wave in first medium,Reflected wave and transmitted wave must have common frequency, but this superpositioned wave IS the incident wave. How do you think the boundary point gets its frequency if not from the incident wave?

• If I start a pulse Using my hand(rigidly) to give initial particle of the rope a Fixed frequency,Then that point will get the frequency That I would like to give,But in this case,instead of my hand,The end particle of rarer medium(which is not rigid) is giving frequency to denser medium's first particle,Won't it affect frequency as soon as superpositioned wave forms,Because way the tension acts(direction) at later part of time is not similiar to the way it should act for particle in incident wave only?Sorry,As I cannot comprehend properly,Please reply If I need to ask question other way! Sep 25, 2022 at 21:52
• It is very hard to understand what you try to say. You have an incident wave, whic is caused farther away. this wave brings a particle at the border to oscillate in the frequency of the wave, Now this frequency continues in both media, one part is reflected and added to the incoming wave, one is continuing in the second medium. How should any other frequency happen? What you mean with "tension which acts later I can not understand. do you talk about waves in general or just waves on a string like your picture? Sep 26, 2022 at 17:05
• ,Though I am still Studying Mechanical waves,So I am not pretty sure for General wave properties(applicable for all Waves),what I was trying to say is,When a pulse is created,As soon as Boundry Particle recieves frequency from incident wave,A shock is given Back(reflected wave+incident wave superposition),which cause change in shape of Wave behind boundry at that Interval of time,Shouldn't it Affect the frequency of motion Of boundry particle,As tension force direction on that boundry particle particle is changed(or is different from normal incident wave)? Sep 26, 2022 at 23:48
• You do not argue with frequency, just with one puls, Try to argument with the frequency of the boundary particle and how you suppose something could oscillate faster or slower. The picture with your puls gives you no information of frequency Sep 27, 2022 at 11:19
• Can't it be said that time required for boundry particle to reach max. Amplitude and come back again at mean position will be it's frequency?(for a pulse) Sep 27, 2022 at 13:32