Sum of the two sound waves

[mgoddard]

Hi, I need to determine the sum of the two sound waves and express it in the form y=Csin(kx-wt-"theta") where w is omega. The two waves are

y=Asin(kx-wt) where w is omega
y=Bsin(kx-wt-"phi") where w is omega

I used a trigometric formula and got it to the point where it equals

y=Asin(kx-wt)+B[sin(kx-wt)cos(phi)-cos(kx-wt)sin(phi)] and I have simplified a little but I cannot get it into the form stated above where y=Csin(kx-wt-theta) Any suggestions?

As well the question states C depends on A,B and phi, and Theta depends on A, B, and phi as well.

Thanks for your time
M Goddard

 

[Galileo]

Hint: Use complex numbers

 

[Architeuthis Dux]

I would approach this problem by first understanding the physical nature of sound waves. Sound waves are mechanical waves that travel through a medium, such as air, and are created by vibrations. These vibrations can be represented by a mathematical function, such as the sine function, which describes the displacement of the particles in the medium.

To determine the sum of two sound waves, we need to understand how they interact with each other. When two waves meet, they combine to create a new wave, known as the superposition of waves. This new wave will have a different amplitude, frequency, and phase compared to the individual waves.

In this case, we have two waves with the same frequency (w) and different amplitudes (A and B), and phases (0 and phi). Using the trigonometric identity, we can rewrite the sum of these two waves as:

y = Asin(kx-wt) + Bsin(kx-wt-phi)

= Asin(kx-wt) + B[sin(kx-wt)cos(phi) - cos(kx-wt)sin(phi)]

= (A+Bcos(phi))sin(kx-wt) - Bsin(phi)cos(kx-wt)

Comparing this to the desired form, we can see that C = √(A² + B² + 2ABcos(phi)) and θ = tan⁻¹(Bsin(phi)/(A+Bcos(phi))).

Therefore, the final expression for the sum of the two sound waves is:

y = √(A² + B² + 2ABcos(phi))sin(kx-wt-tan⁻¹(Bsin(phi)/(A+Bcos(phi))))

In conclusion, by understanding the physical nature of sound waves and using trigonometric identities, we can determine the sum of two sound waves and express it in the desired form.