Speed of a Transverse Wave
Speed of a Transverse Wave
Definition: Wave Speed
Wave speed is the distance a wave travels per unit time.
Quantity: Wave speed (\(v\)) Unit name: metre per second Unit symbol: \(\text{m·s$^{-1}$}\)
The distance between two successive crests is \(\text{1}\) wavelength, \(λ\). Thus in a time of \(\text{1}\) period, the wave will travel \(\text{1}\) wavelength in distance. Thus the speed of the wave, \(v\), is:
\[v = \frac{\text{distance travelled}}{\text{time taken}} = \frac{\lambda}{T}\]However, \(f = \frac{1}{T}\). Therefore, we can also write:
\begin{align*} v & = \frac{\lambda}{T} \\ & = \lambda \cdot \frac{1}{T} \\ & = \lambda \cdot f \end{align*}We call this equation the wave equation. To summarise, we have that \(v = \lambda \cdot f\) where
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\(v =\) speed in \(\text{m·s$^{-1}$}\)
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\(\lambda =\) wavelength in \(\text{m}\)
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\(f =\) frequency in \(\text{Hz}\)
Wave equation:
\[v = f \cdot \lambda\]or
\[v = \frac{\lambda}{T}\]Example:
Question
When a particular string is vibrated at a frequency of \(\text{10}\) \(\text{Hz}\), a transverse wave of wavelength \(\text{0.25}\) \(\text{m}\) is produced. Determine the speed of the wave as it travels along the string.
Step 1: Determine what is given and what is required
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frequency of wave: \(f = \text{10}\text{ Hz}\)
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wavelength of wave: \(\lambda = \text{0.25}\text{ m}\)
We are required to calculate the speed of the wave as it travels along the string.
All quantities are in SI units.
Step 2: Determine how to approach the problem
We know that the speed of a wave is:
\[v = f \cdot \lambda\]and we are given all the necessary quantities.
Step 3: Substituting in the values
\begin{align*} v & = f \cdot \lambda \\ & = (\text{10}\text{ Hz})(\text{0.25}\text{ m}) \\ & = (\text{10}\text{ s$^{-1}$})(\text{0.25}\text{ m}) \\ & = \text{2.5}\text{ m·s$^{-1}$} \end{align*}Step 4: Write the final answer
The wave travels at \(\text{2.5}\) \(\text{m·s$^{-1}$}\) along the string.
Example:
Question
A cork on the surface of a swimming pool bobs up and down once every second on some ripples. The ripples have a wavelength of \(\text{20}\) \(\text{cm}\). If the cork is \(\text{2}\) \(\text{m}\) from the edge of the pool, how long does it take a ripple passing the cork to reach the edge?
Step 1: Determine what is given and what is required
We are given:
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frequency of wave: \(f = \text{1}\text{ Hz}\)
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wavelength of wave: \(\lambda = \text{20}\text{ cm}\)
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distance of cork from edge of pool: \(D = \text{2}\text{ m}\)
We are required to determine the time it takes for a ripple to travel between the cork and the edge of the pool.
The wavelength is not in SI units and should be converted.
Step 2: Determine how to approach the problem
The time taken for the ripple to reach the edge of the pool is obtained from:
\[t = \frac{D}{v} \qquad \left(\text{from } v = \frac{D}{t}\right)\]We know that
\[v = f \cdot \lambda\]Therefore,
\[t = \frac{D}{f \cdot \lambda}\]Step 3: Convert wavelength to SI units
\[\text{20}\text{ cm} = \text{0.2}\text{ m}\]Step 4: Solve the problem
\begin{align*} t & = \frac{D}{f \cdot \lambda} \\ & = \frac{\text{2}\text{ m}}{(\text{1}\text{ Hz})(\text{0.2}\text{ m})} \\ & = \frac{\text{2}\text{ m}}{(\text{1}\text{ s$^{-1}$})(\text{0.2}\text{ m})} \\ & = \text{10}\text{ s} \end{align*}Step 5: Write the final answer
A ripple passing the leaf will take \(\text{10}\) \(\text{s}\) to reach the edge of the pool.
This lesson is part of:
Mechanical Waves and Sound