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# 8.7 Speed of a transverse wave

## 8.7 Speed of a transverse wave (ESACQ)

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

• $$v =$$ speed in $$\text{m·s^{-1}}$$

• $$\lambda =$$ wavelength in $$\text{m}$$

• $$f =$$ frequency in $$\text{Hz}$$

Wave equation:

$v = f \cdot \lambda$

or

$v = \frac{\lambda}{T}$

## Worked example 4: Speed of a transverse wave I

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.

### Determine what is given and what is required

• frequency of wave: $$f = \text{10}\text{ Hz}$$

• 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.

### 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.

### 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*}

The wave travels at $$\text{2,5}$$ $$\text{m·s^{-1}}$$ along the string.

## Worked example 5: Speed of a transverse wave II

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?

### Determine what is given and what is required

We are given:

• frequency of wave: $$f = \text{1}\text{ Hz}$$

• wavelength of wave: $$\lambda = \text{20}\text{ cm}$$

• 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.

### 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,

$t = \frac{D}{f \cdot \lambda}$

### Convert wavelength to SI units

$\text{20}\text{ cm} = \text{0,2}\text{ m}$

### 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*}

A ripple passing the leaf will take $$\text{10}$$ $$\text{s}$$ to reach the edge of the pool.

## Waves

Textbook Exercise 8.1

When the particles of a medium move perpendicular to the direction of the wave motion, the wave is called a             wave.

Solution not yet available

A transverse wave is moving downwards. In what direction do the particles in the medium move?

Solution not yet available

Consider the diagram below and answer the questions that follow:

1. the wavelength of the wave is shown by letter            .

2. the amplitude of the wave is shown by letter            .

Solution not yet available

Draw 2 wavelengths of the following transverse waves on the same graph paper. Label all important values.

1. Wave 1: Amplitude = $$\text{1}$$ $$\text{cm}$$, wavelength = $$\text{3}$$ $$\text{cm}$$

2. Wave 2: Peak to trough distance (vertical) = $$\text{3}$$ $$\text{cm}$$, crest to crest distance (horizontal) = $$\text{5}$$ $$\text{cm}$$

Solution not yet available

You are given the transverse wave below.

Draw the following:

1. A wave with twice the amplitude of the given wave.

2. A wave with half the amplitude of the given wave.

3. A wave travelling at the same speed with twice the frequency of the given wave.

4. A wave travelling at the same speed with half the frequency of the given wave.

5. A wave with twice the wavelength of the given wave.

6. A wave with half the wavelength of the given wave.

7. A wave travelling at the same speed with twice the period of the given wave.

8. A wave travelling at the same speed with half the period of the given wave.

Solution not yet available

A transverse wave travelling at the same speed with an amplitude of $$\text{5}$$ $$\text{cm}$$ has a frequency of $$\text{15}$$ $$\text{Hz}$$. The horizontal distance from a crest to the nearest trough is measured to be $$\text{2,5}$$ $$\text{cm}$$. Find the

1. period of the wave.

2. speed of the wave.

Solution not yet available

A fly flaps its wings back and forth 200 times each second. Calculate the period of a wing flap.

Solution not yet available

As the period of a wave increases, the frequency increases/decreases/does not change.

Solution not yet available

Calculate the frequency of rotation of the second hand on a clock.

Solution not yet available

Microwave ovens produce radiation with a frequency of $$\text{2 450}$$ $$\text{MHz}$$ ($$\text{1}$$ $$\text{MHz}$$ = $$\text{10}^{\text{6}}$$ $$\text{Hz}$$) and a wavelength of $$\text{0,122}$$ $$\text{m}$$. What is the wave speed of the radiation?

Solution not yet available

Study the following diagram and answer the questions:

1. Identify two sets of points that are in phase.

2. Identify two sets of points that are out of phase.

3. Identify any two points that would indicate a wavelength.

Solution not yet available

Tom is fishing from a pier and notices that four wave crests pass by in $$\text{8}$$ $$\text{s}$$ and estimates the distance between two successive crests is $$\text{4}$$ $$\text{m}$$. The timing starts with the first crest and ends with the fourth. Calculate the speed of the wave.

Solution not yet available