Question

A sinusoidal wave travels along a string. The time for a particular point to move from maximum displacement to zero displacement is \(178 \mathrm{~ms}\). The wavelength of the wave is \(1.38\) \(\mathrm{m}\). Find \((a)\) the period, \((b)\) the frequency, and \((c)\) the speed of the wave.

Short answer

The period of the wave is 0.356 seconds, the frequency is 2.81 Hz, and the speed is 3.88 m/s.

Step-by-step solution

Calculation of the period

First, let's find the period of the wave. The time for a particular point to move from maximum displacement to zero displacement is actually half of a full period for a sinusoidal wave. This is because a sinusoidal wave goes from maximum displacement, to zero, to maximum displacement on the opposite side, and back to zero during one full cycle, or period. Therefore, the period (\(T\)) of the wave is twice the given time. So, \(T = 2 \times 178 ms = 356 ms = 0.356 s\).

Calculation of the frequency

Next, the frequency can be found since it is the reciprocal of the period. Hence the frequency (\(f\)) can be calculated by the formula \(f = 1/T\). Substituting in our calculated period of 0.356 seconds yields \(f = 1/0.356 s^{-1} = 2.81 Hz\).

Calculation of the speed

Finally, the speed of the wave can be found using the equation \(v = f \times \lambda\), where \(v\) is the speed, \(f\) is the frequency, and \(\lambda\) is the wavelength. Using our calculated frequency of 2.81 Hz and given the wavelength of 1.38 m, gives \(v = 2.81 Hz \times 1.38 m = 3.88 m/s\).

Key concepts

Understanding Wave Period

The wave period (\( T \)) is the time it takes for one complete cycle of a wave to pass a given point. It is a fundamental property of any wave, including sinusoidal waves that are commonly encountered in physics. In our exercise, given that the time for a peak to reach zero displacement is one quarter of a wave cycle, the period is the time for a full cycle, which is four times that amount.

To make this more relatable, imagine watching a float bobbing up and down on the water. The period would be the time it takes for the float to return to its highest point after completing one full up and down motion. To ensure students fully understand this concept, we focus on the fact that a sinusoidal wave covers a full range from maximum to minimum displacement and back to maximum in one period.

Deciphering Wave Frequency

Wave frequency (\( f \)), measured in hertz (Hz), indicates how many cycles of a wave occur in one second. It's the reciprocal of the period (\( f = 1/T \)). Understanding frequency is similar to counting how many times that float bobs in the water every second. A higher frequency means more cycles per second, and hence more bobbing. When explaining frequency, it's helpful to emphasize its inverse relationship with the period; namely, the shorter the period (faster the cycle), the higher the frequency. With our example problem, we've calculated the frequency to be around 2.81 Hz, meaning that the wave completes approximately 2.81 cycles every second.

Calculating Wave Speed

Wave speed (\( v \)) is a measure of how fast the wave travels through the medium. In physics, this is calculated as the product of frequency and wavelength (\( v = f \times \tau \)), where wavelength (\( \tau \)) is the distance between two corresponding points on consecutive cycles of the wave. To visualize this, if you throw a stone into a still pond, wave speed is how fast those ripples move outward.

It's important for students to recognize that this speed is not how fast an individual particle moves, but how fast the wave pattern moves through the medium. In our problem, with a frequency of 2.81 Hz and a wavelength of 1.38 meters, the wave speed is the product of these two values, resulting in 3.88 meters per second. This value represents the propagation speed of the wave's energy through the string, not the movement of the string itself.