A sinusoidal wave is traveling on a string with speed 40 cm/s. The displacement of the particles of the string at x = 10 cm varies with time according to
y=(5.0cm)sin[1.0−(4.0s−1)t]
. The linear density of the string is 4.0 g/cm. What are (a) the frequency and (b) the wavelength of the wave? If the wave equation is of the form y(x, t) = y_m sin(kx ± ωt), what are (c) y_m, (d) k, (e) ω, and (f ) the correct choice of sign in front of ω? (g) What is the tension in the string?

Given Information:  

Speed = v = 40 cm/s = 0.40 m/s

Linear density of the string = μ = 4.0 g/cm = 0.4 kg/m

Required Information:  

(a) the frequency = ?

(b) the wavelength of the wave = ?

(c) y_m = ?

(d) k = ?

(e) ω = ?

(f) the correct choice of sign in front of ω = ?

(g) What is the tension in the string = ?

(a) the frequency = 2/π Hz

(b) the wavelength of the wave = 0.628 m

(c) y_m = 0.05 m

(d) k = 10 m⁻¹

(e) ω = 4.0 s⁻¹

(f) the correct choice of sign in front of ω = negative sign

(g) What is the tension in the string = 0.064 N.

Explanation:

The standard form of the equation is given as

y(x, t) = y_m sin(kx ± ωt)

Whereas the given equation is

y=(5.0cm)sin[1.0−(4.0s⁻¹)t]

(a) the frequency

comparing the given equation with the standard form yields,

ωt = (4.0s⁻¹)t

ω = 4.0 s⁻¹

We also know that

ω = 2πf

4.0 = 2πf

f = 4.0/2π

f = 2/π Hz

Therefore, frequency of the wave is 2/π Hz

(b) the wavelength of the wave

We know that wavelength is given by

λ = v/f

Where λ is the wavelength, v is the speed of wave and f is the frequency of sinusoidal wave.

λ = 0.40/(2/π)

λ = 0.20π

λ = 0.628 m

(c) y_m

comparing the given equation with the standard form yields,

y_m = 5 cm

or

y_m = 0.05 m

This is the amplitude of the sinusoidal wave

(d) k

comparing the given equation with the standard form yields,

kx = 1

k = 1/x

k = 1/0.10

k = 10 m⁻¹

(e) ω

already calculated in part (a)

ω = 4.0 s⁻¹

(f) the correct choice of sign in front of ω?

comparing the given equation with the standard form, a negative sign is the correct choice of sign in front of ω.

(g) What is the tension in the string?

We know that tension in the string is given by

T = μv²

Where μ is the linear density of the string and v is the speed of the wave.

T = 0.4(0.40)²

T = 0.064 N

Therefore, the tension in the string is 0.064 N.