Question

Consider the following equations for traveling waves on two different strings:

(a) Which wave has the faster wave speed? What is that speed? 

(b) Which wave has the longer wavelength? What is that wavelength? 

(c) Which wave has the faster maximum speed of a point in the medium? What is that speed? 

(d) Which wave is moving in the positive x-direction?

 

 

Answer 1

For first wave A1 = 1.5cm, ω1 = 6s-1, k1 = 4cm-1

For second wave A2 = 4.5 cm, ω2 = -3s-1, k2 = 3cm-1

 

a)

The speed of the wave is given as

v = ω/k

 

For the first wave

v1 = ω1/k1 ⇒ 6s-1/4cm-1

     = 1.5 cm/s

 

For the second wave

v2 = ω2/k2 ⇒ 3s-1/3cm-1

     = 1 cm/s

 

The first wave is moving faster.

 

b)

The wavelength of the wave is given as

λ = 2π/k

 

For the first wave

λ1 = 2π/k1 ⇒ 2π/4cm-1

     = 1.57 cm

 

For the second wave

λ1 = 2π/k1 ⇒ 2π/3cm-1

     = 2.09 cm

 

The second wave has longer wavelength.

 

c)

Maximum speed of appoint in a medium is

v = ωA

 

For the first wave

v1m = ω1A1 ⇒ (6s-1)(1.5cm)

         = 9 cm/s

 

For the second wave

v2m = ω2A2 ⇒ (3s-1)(4.5 cm)

        = 13.5 cm/s

 

The second wave has maximum speed of a point in the medium.

 

d)

For the wave to move in positive x direction then it should in the form of

y(x, t) = A sin(kx – ωt)

 

The second wave is in the same form as in the above so second equation is moving towards positive x-direction.

a. The first wave is moving faster.

b. The second wave has longer wavelength.