Rank the waves according to their wave speed, greatest first. Rank the waves acc

Question

Rank the waves according to their wave speed, greatest first.

Rank the waves according to the tension in the strings along which they travel, greatest first:

The following four waves are sent along strings with the same linear densities (x is in meters and t is in seconds).

1.
2.
3.
4.


In the following questions, you will need to rank the various waves. If multiple waves rank equally, use the same rank for each, then exclude the intermediate ranking (i.e. if objects A, B, and C must be ranked, and A and B must both be ranked first, the ranking would be A:1, B:1, C:3). If all waves rank equally, rank each as '1'.

Explanation / Answer

Wave speed doesn't depend on amplitude, so amplitude values can be ignored.

x = Asin( kx - t) is the basic equation for the displacement of a 1-dimensional wave moving in the positive x-direction. (Cosine can also be used, and a phase angle is sometimes included )
k = 2/ so = 2/k
= 2/T so T = 2/

Since v = f and f=1/T
v= /T
v = (2/k) / (2/)
= /k

Wave 1: k = 1, = 3, v = /k = 3/1 = 3 units
Wave 2: k = 2, = 1, v = /k = 1/2 = 0.5 units
Wave 3: k = 4, = 1, v = /k = 1/4 = 0.25 units
Wave 4: k = 1, = 2, v = /k = 2/1 = 2 units

Speeds, faster to slowest are 1,4,2,3.

Assuming the strings are made of the same material (same thickness and density) wave speed is proportional to the square root of tension.

It follows that the tensions are in the same order, i.e. 1,4,2,3.

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