Wave Help: Solving Standing Wave and Interference Problems

[avs]

I am stuck with these 2 questions. Can someone help. THanks



1. A string is tied tightly between two fixed points 0.7 meter apart (along the x-axis) so that its tension is 9N. A 3 meter length of the same string has a mass of 18g. Very small oscillations of one of the fixed points at a carefully chosen frequency, cause a corresponding transverse standing wave mode to be set up. Assume the oscillations are small enough that both sides of the strings are nodes. The modes are set up one at a time. In all cases the maximum displacement of any part of the string is 2.0mm from the x-axis and lies in the y-direction.

a. Evaluate the two lowest standing wave frequencies, f.
b. Sketch the standing waves corresponding to the two frequencies in (a) labeling each with its correct frequency.
c. Determine the maximum accelerations and velocities (vector!) that exist in the above standing waves.
d. For each of the answers to part ( C ), show on a sketch where the string is in its cycle and where on the string each maximum is to be found.


2. Consider two points sources located at (x1=3 cm, y1=z1=0) and (x2=-3cm, y2=z2=0) respectively. They are emitting identical waves which spread out equally in all directions (spherical wave fronts). The wavelength is 2.3 cm. The two sources are oscillating in phase with each other.

a. In the x-y plane at large distances from the sources, find all the angles from the y-axis at which you would find constructive interference.
b. Similarily find all the angles at which you would find destructive interference.
c. Bearing in mind the three dimensional character of the problem, sketch a perspective view showing where destructive interference occurs.
d. Suppose now that the two sources are exactly pi/2 radians out of phase with each other. Again find the angles at which destructive interference and again sketch a perspective view.

 

[frankR]

What aren't you understanding and please show us where you are getting stuck.

Thanks

 

[avs]

Ok just want to confirm if 1a is correct.
thanks a lot!

First K1=(pi/L)=4.5m^-1 K2=(2*pi)/L=9.0m^-1
Now Lambda(1)=(2*pi)/K1=1.4m Lambda(2)=(2*pi)/K2=0.7m

f*lambda=(T/mu)^1/2 mu=0.018kg/3m=0.006kg/m
f1=27.7 f2=55.33

 

[quantumdude]

Yes, you did that correctly.

 

[avs]

ok for 1c:
1st node:
v=f*(lambda) lambda=2L=1.4m
V=27.7*1.4m=38.8m/s

2nd node:
lambda=L
V=55.33*0.7=38.7m/s

For acceleration:
V=f*lambda=(lambda/tau)
dV/dTau=(-lambda/tau^2)
tau=1/f

plug everything in i got
node1:
a=1.4/0.0361^2=1074.2m/s^2

node2:
a=0.7/0.01807^2=2143m/s^2

Just want to confirm if this is correct. Thanks

 

[quantumdude]

Originally posted by avs
ok for 1c:
1st node:
v=f*(lambda) lambda=2L=1.4m
V=27.7*1.4m=38.8m/s

2nd node:
lambda=L
V=55.33*0.7=38.7m/s

First, you aren't evaulating the velocity at any "node". A node is a point that does not move at all.

Second, you are computing the wave velocity, and that is not what they are asking for. They are asking for the transverse velocity, the velocity at which a point moves up and down.

You'll need to start with the equation for the displacement of a standing wave:

y(x,t)=y_msin(kx±ωt).

For acceleration:
V=f*lambda=(lambda/tau)
dV/dTau=(-lambda/tau^2)
tau=1/f

plug everything in i got
node1:
a=1.4/0.0361^2=1074.2m/s^2

node2:
a=0.7/0.01807^2=2143m/s^2

Since you started from the wrong point, this is wrong too. You'll need to work it from y(x,t), given above.