Wave Equations

Wavelength is the physical distance from one point on a wave to the exact same point on the next wave (for example, from peak to peak). You can calculate wave speed by multiplying these two values together:

$$v = f \times \lambda$$

Where:

• $v$ = wave speed in metres per second ($\text{m/s}$)
• $f$ = frequency in Hertz ($\text{Hz}$)
• $\lambda$ = wavelength in metres ($\text{m}$)

Worked Example: Radio Station Calculation A radio station broadcasts at $200 \text{ kHz}$ with a wavelength of $1500 \text{ m}$. Calculate the wave speed.

Step 1: Convert units.

• $200 \text{ kHz} = 200,000 \text{ Hz}$

Step 2: State the formula.

• $v = f \times \lambda$

Step 3: Substitute the values.

• $v = 200,000 \times 1500$

Step 4: Calculate the final answer.

• $v = 300,000,000 \text{ m/s}$ (or $3 \times 10^8 \text{ m/s}$)

Calculating Speed using Distance and Time

All electromagnetic waves travel at a constant speed of $3.0 \times 10^8 \text{ m/s}$ in a vacuum. Sound travels much slower, at roughly $330 \text{ m/s}$ to $340 \text{ m/s}$ in air, and approximately $1500 \text{ m/s}$ in water.

If you know the distance a wave has travelled and the time it took, you can calculate its speed. Note that the Edexcel exam board specifically uses the symbol $x$ for distance:

$$v = \frac{x}{t}$$

Where:

• $v$ = wave speed in metres per second ($\text{m/s}$)
• $x$ = distance in metres ($\text{m}$)
• $t$ = time in seconds ($\text{s}$)

Worked Example: Lightning Distance Thunder is heard $4 \text{ s}$ after a lightning flash. If the speed of sound is $330 \text{ m/s}$, how far away is the storm?

Step 1: Rearrange the formula for distance ($x$).

• $x = v \times t$

Step 2: Substitute the values.

• $x = 330 \times 4$

Step 3: Calculate the final distance.

• $x = 1320 \text{ m}$

Echoes and Average Speed

In experiments, you may need to calculate the average speed of a wave over a distance. For reflection scenarios like sonar or echoes, you must account for the wave travelling "there and back."

Worked Example: Speed of Sound via Echo A student stands $220 \text{ m}$ from a wall. They hear an echo $1.33 \text{ s}$ after clapping two wooden blocks together. Calculate the speed of sound.

Step 1: Calculate total distance ($x$).

• The sound travels to the wall and back: $x = 220 \times 2 = 440 \text{ m}$

Step 2: Substitute into the speed formula.

• $v = \frac{440}{1.33}$

Step 3: Calculate the final speed.

• $v = 330.8 \text{ m/s}$ (or $331 \text{ m/s}$ to 3 significant figures)

Crucial Unit Conversions

Before using either wave equation, you must ensure your units are standard ($\text{m/s}$, $\text{Hz}$, $\text{m}$, and $\text{s}$). Be prepared to convert the following prefixes:

• Kilohertz (kHz): Multiply by $1,000$ ($\times 10^3$) to get $\text{Hz}$.
• Megahertz (MHz): Multiply by $1,000,000$ ($\times 10^6$) to get $\text{Hz}$.
• Gigahertz (GHz): Multiply by $1,000,000,000$ ($\times 10^9$) to get $\text{Hz}$.
• Nanometre (nm): Multiply by $10^{-9}$ to get $\text{m}$.