Question

To see how two traveling waves of the same frequency create a standing wave.

Consider a traveling wave described by the formula

y1(x,t)=Asin(kx−ωt).

This function might represent the lateral displacement of a string, a local electric field, the position of the surface of a body of water, or any of a number of other physical manifestations of waves.

a)

Part A

Part complete

Which one of the following statements about the wave described in the problem introduction is correct?

The wave is traveling in the +x direction.

The wave is traveling in the −x direction.

The wave is oscillating but not traveling.

The wave is traveling but not oscillating.

b)

Which of the expressions given is a mathematical expression for a wave of the same amplitude that is traveling in the opposite direction? At time t=0this new wave should have the same displacement as y1(x,t), the wave described in the problem introduction.

Acos(kx−ωt)

Acos(kx+ωt)

Asin(kx−ωt)

Asin(kx+ωt)

The principle of superposition states that if two functions each separately satisfy the wave equation, then the sum (or difference) also satisfies the wave equation. This principle follows from the fact that every term in the wave equation is linear in the amplitude of the wave.

Consider the sum of two waves y1(x,t)+y2(x,t), where y1(x,t) is the wave described in Part A and y2(x,t) is the wave described in Part B. These waves have been chosen so that their sum can be written as follows:

ys(x,t)=ye(x)yt(t).

This form is significant because ye(x), called the envelope, depends only on position, and yt(t) depends only on time. Traditionally, the time function is taken to be a trigonometric function with unit amplitude; that is, the overall amplitude of the wave is written as part of ye(x).

Part C

Find ye(x) and yt(t). Keep in mind that yt(t) should be a trigonometric function of unit amplitude.

Express your answers in terms of A, k, x, ω (Greek letter omega), and t. Separate the two functions with a comma.

d)

At the position x=0, what is the displacement of the string (assuming that the standing wave ys(x,t) is present)?

Express your answer in terms of parameters given in the problem introduction.

Part F

At certain times, the string will be perfectly straight. Find the first time t1>0 when this is true.

Express t1 in terms of ω, k, and necessary constants.

Answer 1

Part A:

(a) The principle of superposition follows that every term in the wave equation is linear in the amplitude of the wave.

Therefore, The wave is travelling in the +x direction

(b) As the waves are same and directions are opposite, the signs will also be opposite.

Part C:

The sum of two waves were,   

= y₁(x,t) (the waves described in part A)

=y₂(x₂,t) (the waves described in part B)

Now their sum can be written as follows,

y_s(x,t) = y_e(x) y_t(t)

So we now have to find the sum of the equations in part (a) and part (b),

On applying the trigonometric identity,

Sin(A-B) = Sin(A)Cos(B) - Cos(A)Sin(B)

Which will give,

(d) y_s(x=0,t) = 0

Part (F),

The equation of standing wave in part C is,

The string can only be straight when

and for then y(x,t)=0 also (for all x)

For any other value , it will be sinusoidal.

If ,