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

Consider a traveling wave described by the formula

y1(x,t)=Asin(kx−ωt).

This function might represent the lateral displacement of astring, a local electric field, the position of the surface of abody of water, or any of a number of other physical manifestationsof waves.

a)

Part A

Part complete

Which one of the following statements about the wave describedin 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 fora wave of the same amplitude that is traveling in the oppositedirection? At time t=0this new wave should have the samedisplacement as y1(x,t), the wave described in the problemintroduction.

Acos(kx−ωt)

Acos(kx+ωt)

Asin(kx−ωt)

Asin(kx+ωt)

The principle of superposition states that if twofunctions each separately satisfy the wave equation, then the sum(or difference) also satisfies the wave equation. This principlefollows from the fact that every term in the wave equation islinear in the amplitude of the wave.

Consider the sum of two waves y1(x,t)+y2(x,t), where y1(x,t) isthe wave described in Part A and y2(x,t) is the wave described inPart B. These waves have been chosen so that their sum can bewritten 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 trigonometricfunction with unit amplitude; that is, the overall amplitude of thewave is written as part of ye(x).

Part C

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

Express your answers in terms of A, k, x, ω (Greek letteromega), 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 problemintroduction.

Part F

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

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