Question

A wave along a string has the following equation \( y = 0.02 \sin [30t – 4.0x] \) m. The speed of the wave is:

• A. 4.0 m/s
• B. 30 m/s
• C. 7.5 m/s
• D. 10 m/s

Hint:

The speed of a wave on a string is the ratio of angular frequency to the wave number.

Answer 1

The Correct Option is C

The given wave equation is \( y = 0.02 \sin [30t - 4.0x] \) m.

To find the speed of the wave, we use the general form of the wave equation: \( y = A \sin ( \omega t - kx ) \), where:

• \( \omega \) is the angular frequency
• \( k \) is the wave number
• The wave speed \( v \) is given by: \( v = \frac{\omega}{k} \)

From the given equation:

• \( \omega = 30 \, \text{rad/s} \)
• \( k = 4.0 \, \text{rad/m} \) 

Substitute these values into the formula for wave speed:

\( v = \frac{30}{4.0} = 7.5 \, \text{m/s} \)

Thus, the speed of the wave is 7.5 m/s.

Answer 2

The general form of the wave equation is: \[ y = A \sin(kx - \omega t) \] where:
- \( A \) is the amplitude,
- \( k = 4.0 \, \text{rad/m} \) is the wave number,
- \( \omega = 30 \, \text{rad/s} \) is the angular frequency. The wave speed \( v \) is given by: \[ v = \frac{\omega}{k} \] Substituting the given values: \[ v = \frac{30}{4.0} = 7.5 \, \text{m/s} \] Thus, the speed of the wave is 7.5 m/s.

Answer 3

For a wave \( y = A\sin(\omega t - kx) \), points of constant phase satisfy \( \omega t - kx = \text{constant} \). Differentiating this condition with respect to time gives the wave speed directly: \( \omega - k\dfrac{dx}{dt} = 0 \), so \( \dfrac{dx}{dt} = \dfrac{\omega}{k} \). Here \( \omega = 30\,\text{rad/s} \) and \( k = 4.0\,\text{rad/m} \), so \( \dfrac{dx}{dt} = \dfrac{30}{4.0} = 7.5\,\text{m/s} \). Let's see how each option holds up against this.

1. 4.0 m/s: This is actually the value of \(k\), the wave number, not the speed; mistaking the wave number for the speed gives this wrong option.
2. 30 m/s: This is the angular frequency \(\omega\) taken on its own, without dividing by \(k\); using \(\omega\) alone leads here.
3. 7.5 m/s: Differentiating the constant-phase condition gives \( dx/dt = \omega/k = 30/4.0 = 7.5\,\text{m/s} \) exactly, so this matches the wave speed found above.
4. 10 m/s: This does not follow from \(\omega/k\) or any other simple combination of the given numbers, so it does not satisfy the phase condition.

Since the wave speed is fixed by how fast a point of constant phase moves, and that works out to \(\omega/k = 7.5\,\text{m/s}\), the third option is the one consistent with the equation.

So the correct answer is 7.5 m/s.