How to Find the Resultant Wave Equation of Two Plane Waves?

[thebigstar25]

Homework Statement

I have two plane waves, one of the form: 4 sin(20t + (pi/3)x + pi), and the other one: 2 sin(20t +(pi/4)y + pi) .. with the same frequency and vibrations in the z direction ..

I am asked to find the resultant wave equation at x = 5 and y =2

Homework Equations

maybe x = r costheta

The Attempt at a Solution

I have the answer to this question as 2.48sin(20t-0.2pi), I know I can't just substitute the x and y in the waves and simply add them .. and unfortunately the book doesn't discuss such situation .. I would appreciate any hints and suggestions .. Thanks in advance

 

[tiny-tim]

hi thebigstar25!

(have a pi: π )

I have the answer to this question as 2.48sin(20t-0.2pi), I know I can't just substitute the x and y in the waves and simply add them …

yes you can …

substitute the x and y, and use sin(A+ B) = sinAcosB + cosAsinB

 

[thebigstar25]

I did that and I got -2sin20t + 1.46cos20t which doesn't look like the final answer?

 

[tiny-tim]

ok now use the sin(A+ B) = sinAcosB + cosAsinB formula again

(hopefully, B will turn out to be -0.2 π )

 

[thebigstar25]

I am sorry but I can't see where I can apply the formula again

 

[tiny-tim]

A is 20°, B is what you have to find, and tanB = -1.46/2

 

[thebigstar25]

as simple as this problem may seem, I feel like I am missing few things!

first I am not sure why do I have to substitute the values of x and y at the beginning then add the two waves, it would not make a difference if the problem has two waves in the x direction or y direction or one in the x and the other in the y?


my second point is why you suggested to introduce tanB = - 1.46/2 .. I am sorry but I just can't get it :(

 

[tiny-tim]

first I am not sure why do I have to substitute the values of x and y at the beginning then add the two waves, it would not make a difference if the problem has two waves in the x direction or y direction or one in the x and the other in the y?

uhh? but it is one in the x and the other in the y

my second point is why you suggested to introduce tanB = - 1.46/2

because to get sinAcosB + cosAsinB looking like -2sin20t + 1.46cos20t, you need sinB/cosB = 1.46/-2

 

[thebigstar25]

I was just saying that my two waves are as follows:
4 sin(20t + (pi/3)x + pi)
2 sin(20t +(pi/4)y + pi)

I was wondering what if the second wave was 2 sin(20t +(pi/4)x+ pi) with x = 2, then I will end up with the same answer? why is that?


Thanks a lot for the clarification, I will try again ..

 

[thebigstar25]

Thaaaaaaaaaaaaaaaaaaaaaaaanks alooooooot I got the answer .. really I can't thank you enough for your help ! ..

If it is possible to answer my last question in the previous post you will make my day more than great :) .. Thanks again

 

[tiny-tim]

I was wondering what if the second wave was 2 sin(20t +(pi/4)x+ pi) with x = 2, then I will end up with the same answer? why is that?

no, the two waves have to be measured at the same point, you can't measure the first wave at (5,2) and the second at (2, something)

but if you changed the second wave to to 2 sin(20t +(π/10)x+ pi), and measured both at (5,anything), yes you'd get the same result

 

[thebigstar25]

oh! I got it now, that part was confusing me :) .. Thanks a lot again for the great help