# Progressive Waves

## Progressive Waves Revision

**Progressive Waves**

You are likely to be familiar with **progressive waves** from GCSE Physics. A **progressive wave** transfers **energy** from one place to another through a **medium**, without transferring the medium itself.

**Properties of a Progressive Wave**

Here is a diagram of a **progressive wave**. There are several features of a progressive wave you are expected to identify and describe.

**Amplitude** – this is the maximum vertical displacement between the **equilibrium position** and the peak of the wave. We would measure this in metres.

**Wavelength** – this is the horizontal displacement between two identical points on the wave. On the diagram above this is represented between two peaks. Again, this is measured in metres.

**Equilibrium position** – the point at which the wave would form if there was zero **amplitude**.

**Time period** – this is the time taken, in seconds, for the whole **wavelength** to pass a certain point.

**Frequency** – the **frequency** of a wave is related to its **time period**. The **frequency** is the number of waves that pass a given point in one second. Its units are** h****ertz** \text{Hz}, where 1 \text{ Hz} is one wave per second.

To change between **time period** and **frequency**, the following equation can be used:

f=\dfrac{1}{T}

- f=
**frequency**in hertz \text{Hz} - T= the
**time period**in seconds \text{s}.

**The Wave Speed Equation**

This is another equation you will have come across at GCSE level but is still useful in A level physics.

v=f \lambda

- v= the
**wave velocity**in metres per second (\text{ms}^{-1}) - f= the
**frequency**in hertz (\text{Hz}) - \lambda= the
**wavelength**in metres (\text{m}).

It is important to note that if we are studying any **electromagnetic waves **that they all travel at the speed of light and therefore, the equation can be rearranged to become:

c=f\lambda

- c= the
**speed of light**=3\times 10^8 \text{ ms}^{-1}.

**Phase Difference**

Waves from a source may not always reach an object at the same time. The **phase difference** of two waves is a measure of how far two identical points are away from each other on two identical waves.

Here we can see two identical waves, one in red one in blue to help compare. They have the same **wavelength** and **frequency** but the red wave is slightly behind the blue wave. We say they have a **phase difference**.

In the diagram, the blue wave is \dfrac{1}{4} of a **wavelength **(\dfrac{1}{4} \lambda ) ahead of the red wave. This can also be expressed in terms of **degrees** or **radians**.

To convert to **degrees**, we multiply the fraction of the **wavelength** by 360 degrees. So, for the example in the diagram::

\dfrac{1}{4} \times 360 = \bold{90 \degree}

So the two waves are **out of phase** by 90 degrees.

Alternatively, we can convert to radians by multiplying the fraction of the wavelength by 2 \pi. For the example in the diagram this is:

\dfrac{1}{4} \times 2 \pi = \bold{\dfrac{\pi}{2}} \textbf{rad}

If two waves are exactly 360 \degree \text{ or } 2\pi \text{ rad} (or multiples of this) out of phase, then the peaks and troughs of the waves will line up. In this case we say that the two waves are **in phase**.

If the waves are exactly 180 \degree \text{ or } \pi \text{ rad} out of phase, then the two waves are said to be in **anti-phase**.

**Example 1: Calculating Time Period of a Wave**

The frequency of red light is 480 \text{ THz}. What is the time period of a wave of red light?

**[1 mark]**

\begin{aligned} T &=\dfrac{1}{f} \\ &= \dfrac{1}{\textcolor{d11149}{480 \times 10^{12}}} \\ &= \bold{2.1 \times 10^{-15}} \textbf{ s} \end{aligned}

**Example 2: Calculating Wavelength Using the Wave Equation**

Calculate the wavelength of a wave of red light with frequency 480 \text{ THz}.

**[2 marks]**

Light is an EM wave therefore we use:

c=f \lambda

Rearrage to make wavelength the subject:

\begin{aligned} \lambda &= \dfrac{c}{f} \\ &= \dfrac{3 \times 10^8}{\textcolor{ffad05}{480 \times 10^{12}}} \\ &= \bold{6.25 \times 10^{-7}} \textbf{ m or } \bold{625} \textbf{ nm} \end{aligned}

## Progressive Waves Example Questions

**Question 1: **A wave has a measured time period of 0.01 \text{ s}. Calculate its frequency.

**[1 mark]**

\begin{aligned} f&=\dfrac{1}{T} \\ &= \dfrac{1}{0.01} \\ \\ &= \bold{100} \textbf{Hz} \end{aligned}

**Question 2:** The above wave from Question 1 has a wavelength of 400 \text{ nm}. Calculate the speed of the wave.

**[1 mark]**

\begin{aligned} v &=f \lambda \\ &= 100 \times 400 \times 10^{-9} \\ &= \bold{4 \times 10^{-5}} \textbf{ ms} \bold{^{-1}} \end{aligned}

**Question 3:** Two waves are out of phase by \dfrac{1}{3} of a wavelength. What is their phase difference in degrees and radians?

**[2 marks]**

\dfrac{1}{3} \times 360 \degree = \bold{120 \degree} \\ \dfrac{1}{3} \times 2 \pi = \bold{ \dfrac{2}{3} \pi}

**Question 4: **What is a progressive wave?

**[2 marks]**

A progressive wave is wave which transfers **energy from one place to another.**

**W****ithout transfer in matter.**

**Question 5:** Describe what is meant by the terms, in-phase and in anti-phase.

**[2 marks]**

If two waves are in-phase, the **peaks of each of the waves line up at the same point**.

If two waves are in anti-phase, the **peaks and troughs align with each other**. They are exactly half a wave cycle out from one another.