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## 9.5 Speed of a longitudinal wave (ESACW)

The speed of a longitudinal wave is defined in the same was as the speed of transverse waves:

Wave speed

Wave speed is the distance a wave travels per unit time.

Quantity: Wave speed (v)         Unit name: speed         Unit: $$\text{m·s^{−1}}$$

The distance between two successive compressions is 1 wavelength, $$λ$$. Thus in a time of 1 period, the wave will travel 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 $$\text{Hz}$$

## Worked example 1: Speed of longitudinal waves

The musical note “A” is a sound wave. The note has a frequency of $$\text{440}$$ $$\text{Hz}$$ and a wavelength of $$\text{0,784}$$ $$\text{m}$$. Calculate the speed of the musical note.

### Determine what is given and what is required

Using:

\begin{align*} f & = \text{440}\text{ Hz} \\ \lambda & = \text{0,784}\text{ m} \end{align*}

We need to calculate the speed of the musical note “A”.

### Determine how to approach based on what is given

We are given the frequency and wavelength of the note. We can therefore use:

$v = \lambda \cdot f$

### Calculate the wave speed

\begin{align*} v & = f \cdot \lambda \\ & = (\text{440}\text{ Hz})(\text{0,784}\text{ m}) \\ & = \text{345}\text{ m·s$^{-1}$} \end{align*}

The musical note “A” travels at $$\text{345}$$ $$\text{m·s^{−1}}$$.

## Worked example 2: Speed of longitudinal waves

A longitudinal wave travels into a medium in which its speed increases. How does this affect its... (write only increases, decreases, stays the same).

1. period?

2. wavelength?

### Determine what is required

We need to determine how the period and wavelength of a longitudinal wave change when its speed increases.

### Determine how to approach based on what is given

We need to find the link between period, wavelength and wave speed.

### Discuss how the period changes

We know that the frequency of a longitudinal wave is dependent on the frequency of the vibrations that lead to the creation of the longitudinal wave. Therefore, the frequency is always unchanged, irrespective of any changes in speed. Since the period is the inverse of the frequency, the period remains the same.

### Discuss how the wavelength changes

The frequency remains unchanged. According to the wave equation

$v = f \cdot \lambda$

if $$f$$ remains the same and $$v$$ increases, then $$λ$$, the wavelength, must also increase.