How Does Phase Difference Influence the Amplitude of Combined Wave Functions?

[ColdFusion85]

I am not looking for any answers, just some guidance.

Consider case (c), (case (c) involved two waves with equal wavelength and amplitude, but with some arbitrary phase difference), and write the two waves as

$$y1(x) = Acos((\frac{2*\pi*x}{\lambda}))$$
$$y2(x) = Acos((\frac{2*\pi*x}{\lambda}) + \phi)$$

where $$\lambda$$ and $$A$$ are the common wavelength and amplitude of the two waves and $$\phi$$ is their phase difference. Calculate the sum wave $$y(x) = y1(x) + y2(x)$$ and find an expression for the amplitude of the sum wave in terms of $$A$$ and $$\phi$$.

Find an expression for Amplitude in terms of the amplitude?? What exactly is this question asking?

 

[Nate810]

The question is asking you to calculate the amplitude of the sum wave given by the expression y(x) = y1(x) + y2(x). To do this, you will need to use the two given equations for y1(x) and y2(x), as well as the parameters A, λ, and ϕ. After calculating the sum wave, you can then use the equation for the amplitude of a wave (A = √(P/ρ)) to find an expression for the amplitude in terms of A and ϕ.

 

[quicknote]

The question is asking for an expression that relates the amplitude of the resulting sum wave to the amplitude and phase difference of the two individual waves. This can be derived by using the trigonometric identity for the sum of two cosine functions. By substituting the given equations for y1(x) and y2(x) into the sum wave equation, we get:

y(x) = Acos((\frac{2*\pi*x}{\lambda})) + Acos((\frac{2*\pi*x}{\lambda}) + \phi)

Using the trigonometric identity, cos(a) + cos(b) = 2cos(\frac{a+b}{2})cos(\frac{a-b}{2}), we can simplify the expression for y(x) to:

y(x) = 2Acos((\frac{2*\pi*x}{\lambda} + \frac{\phi}{2}))cos(\frac{\phi}{2})

This shows that the amplitude of the resulting sum wave is 2Acos(\frac{\phi}{2}), which is dependent on the amplitude A and the phase difference \phi of the two individual waves. This expression can be further simplified to Amplitude = 2Acos(\frac{\phi}{2}) = 2Acos(\frac{\pi*\phi}{\pi}) = 2Acos(\frac{\pi*\phi}{2\pi}), which gives us a final expression for the amplitude of the sum wave in terms of A and \phi.