Prof that it is not possible to get a standing wave.

[center o bass]

Homework Statement

As in the Title starting from two planewaves with the same amplitude, but different frequency.

Homework Equations

Starting from
$$Ae^{i(kx +\omega_1 t)} + Ae^{i(kx +\omega_2 )}$$

The Attempt at a Solution

I got as far as

$$Ae^{i(kx +\omega_1 t)} + Ae^{i(kx +\omega_2 )}$$
$$= 2Ae^{ikx} e^{i \frac{\omega_2 + \omega_1}{2} t} \cos \frac{\omega_2 - \omega_1}2 t$$

taking the real part

$$= 2A \left( \cos kx \cos{ \frac{\omega_2 + \omega_1}{2} }t - \sin kx \sin { \frac{\omega_2 + \omega_1}{2} } \right) \cos{ \frac{\omega_2 - \omega_1}{2}}$$

Now how do i argue to conclude that this is definitley not a standing wave? I know that position and time has to be decoupled, but how can I conclude that they are not?

Can this expression be simplified further?

is it enough to say that this is a product of two factors one which depends only on time and one which does depend on both space and time and therefore can not be decoupled?

 

[kuruman]

I would start with an expression that reduces to a standing wave when the frequencies are the same. Does your starting expression do that?

 

[center o bass]

Yes it does when $$\omega_1 = - \omega_2$$

 

[kuruman]

By your logic, if ω₁ = 100 Hz, then ω₂ = -100 Hz. Is 100 Hz equal to -100 Hz? Do it right and put the minus sign where it belongs. Wave frequencies are always positive.

 

[paranormal]

As a scientist, it is important to approach this problem with a critical and analytical mindset. First, let's define what a standing wave is. A standing wave is a type of wave that occurs when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. This results in a pattern of nodes and antinodes that appear to be standing still.

Now, let's look at the expression you have derived. It is a product of two factors, one depending on time and the other depending on both space and time. This means that the amplitude at any given point is not constant, but rather varies with time. In order for a standing wave to occur, the amplitude must remain constant at any given point. Therefore, this expression cannot represent a standing wave.

Furthermore, the cosine term in the expression represents a traveling wave, as it varies with both space and time. In a standing wave, the amplitude should not vary with space, only with time. Therefore, this expression does not meet the criteria for a standing wave.

In conclusion, based on the definition of a standing wave and the analysis of the expression provided, it is not possible to get a standing wave from two plane waves with the same amplitude but different frequencies. This is because the resulting expression does not meet the criteria for a standing wave.