A transverse wave on a string is described with the wave function y (x, t) = (0.59 cm) sin[ (1.80 m-1) x - (6.00 s-1) t]. (a) What is the wave velocity? (Enter the magnitude in m/s.) m/s (b) What is the magnitude of the maximum velocity (in m/s) of the string perpendicular to the direction of the motion? m/s

(a) wave velocity = 3.33 m/s².

(b) magnitude of the maximum velocity = 3.54 m/s²

Explanation:

The general equation of a traveling wave is given,

y = Asin (ωt - kx) ... (equation 1)

Where A = Amplitude of the wave (m)

ω = Angular frequency (s⁻¹)

k = Angular wave number (m⁻¹)

(a).

From the expression above, v = ω/k

Given : y = (0.59) sin[ (1.80) x - (6.00) t ... (equation 2)

Comparing Equation 1 and equation 2

1.8x = - kx

∴ k = - 1.8 m⁻¹

And - 6.00t = ωt

∴ ω = - 6.00 s⁻¹

∴ v = - 6.00/-1.8 = 3.33 m/s².

wave velocity = 3.33 m/s².

(b).

We differentiate (equation 2) with respect to time (t) to get an expression for the transverse speed of the wave.

∴ dy/dt = d{ (0.59) sin[ (1.80) x - (6.00) t]/dt

dy/dt = 0.59 (-6.00) cos (1.8x - 6.00t)

dy/dx = 0.59 (-6.00) cos (1.8x - 6.00t)

The magnitude of the maximum velocity = The absolute value of the coefficient of the cosine function.

Vmax = 0.59 * 6.00 = 3.54 m/s²

∴ magnitude of the maximum velocity = 3.54 m/s²