# Stationary Waves

## Stationary Waves Revision

**Stationary Waves**

When a wave is reflected forming two identical waves travelling in opposite directions, a **stationary wave **(often called a** standing wave**) is formed.

**Formation of a Stationary Wave**

The most common way a **stationary wave** is formed between two fixed ends. An example of this is a string fixed at both ends, such as a guitar string.

For a **stationary wave **to form, the waves travelling in each direction must be of the same **frequency**, **wavelength** and **amplitude**.

A **stationary wave **stores **energy** unlike a progressive wave.

In the above example, a **stationary wave** is formed by a full **wavelength** in the forwards direction, followed by a full **wavelength** in the reflected direction. Both directions have the same **amplitude**, **frequency** and **wavelength**. We can also see that **energy** is stored within the system as it has no place to exit as the wave is bound between fixed points at each end of the wave.

**Nodes and Antinodes**

These two terms are used a lot when discussing **stationary waves**, especially in written questions. They describe specific points on a wave.

A **node** is a point on a **stationary wave** where there is zero **amplitude**.

An **anti-node** is a point on a **stationary wave **where there is maximum **amplitude**.

In a **stationary wave **with two closed ends (like the string above), the stationary wave will always have one more **node** than **anti-node**.

To produce a **stationary wave** in a string, the string must be held under **tension**. The equipment below may be used to keep the string under **tension**. It can also be used to investigate the effects of **tension** on the **stationary wave** produced.

The **oscillator** can be set at different **frequencies** until a **stationary wave** is formed. The type of string, **tension** on the string and **length** between the two fixed points can be changed to create a variety of different investigations.

It’s not just waves on a string that form **stationary waves**. Other examples include:

**Microwaves**

A similar experiment to the one above can be completed for **microwaves**. A **microwave** transmitter is used to emit **microwaves** at different **frequencies**. The microwave source is directed towards a shiny metal surface which acts as a **reflector** to reflect the **microwaves** back towards the transmitter. Finally, a microwave detector can be used to find the areas between the reflector and the source where there are **nodes** and **anti-nodes** to allow the waves to be sketched.

**Sound waves **

Some musical instruments use sound to form **stationary waves**. These waves form inside a tube like that of a recorder or clarinet. If sound passes along the tube at the right **frequencies**, **stationary waves** form along the tube.

Because **sound waves **are not visible to the eye, a fine powder can be inserted into a tube in order to visualise the nodes and anti-nodes. When a **stationary wave** is formed, the powder moves into evenly distributed piles. These piles represent the **nodes** of the **stationary wave**. The **frequency** of the sound being produced can be adjusted to investigate how the position and number of piles changes. As the frequency increases, the number of nodes increases and the piles of powder move closer together.

**The First Harmonic**

**Harmonics** form along a wave when **stationary waves** are formed. The easiest way to observe this is along a string under **tension** but **harmonics** form in any **stationary wave**.

The **first harmonic** forms in a **stationary wave** where the wave has two **nodes** and one **anti-node**. This is the **first harmonic **or the simplest **stationary wave** formed. This **harmonic** will form at specific **frequencies** depending upon the **material**, **length** and **tension** of the string.

For the **first harmonic**, if the **length** of the string is L, the **wavelength** would be 2L as the wave needs to reach the end of the string and reflect back again. The **first harmonic** is formed at the **lowest frequency** and as the frequency is increased, other **harmonics** are formed.

To calculate the **frequency** of the **first harmonic** we can use the wave speed equation, replacing the **wavelength** with \bold{2L}:

f = \dfrac{v}{2L}

- f= the
**frequency**in hertz \text{(Hz)} - v= the
**wave speed**in metres per second \text{(ms}^{-1}) - L= the
**length**of the string in metres \text{(m)}.

However, this equation is not the most useful on its own as it is hard to determine the speed of a wave along a string. Instead, we can combine this with another equation:

v=\sqrt{\dfrac{T}{\mu}}

- T= the
**tension**in the string in newtons \text{(N)} - \mu = the
**mass per unit length**of the wire in kilograms per metre \text{(kgm}^{-1})

Substituting v into our original equation gives us:

f=\dfrac{1}{2L} \times \sqrt{\dfrac{T}{\mu}}

This is a much more usable equation as **tension** and **mass per unit length **are easily measured.

**Other Harmonics**

**Harmonics** form whenever a whole number of **stationary waves** form along the string. In the **first harmonic** one full wave is formed. We can use the general equation below to calculate the **frequency** of any **harmonic:**

f=\dfrac{1}{\lambda} \times \sqrt{\dfrac{T}{\mu}}

The** second harmonic**:

This produces \bold{3} **nodes** and \bold{2} **anti-nodes**. The **wavelength** is now \dfrac{2L}{2} which simplifies to be L, and therefore the equation for the **second harmonic** is:

f=\dfrac{1}{L} \times \sqrt{\dfrac{T}{\mu}}

The **third harmonic** forms \bold{4} **nodes** and \bold{3} **anti-nodes**:

The **wavelength** is now \dfrac{2L}{3} and the equation for **frequency** becomes:

f=\dfrac{3}{2L} \times \sqrt{\dfrac{T}{\mu}}

You can now see a pattern between the **harmonic** number, **wavelength** and the number of **nodes** and **anti-nodes**.

**Required Practical 1**

**Determining the frequency of the first harmonic on a string**

Aim: To identify how the** frequency** of the **first harmonic** of a **standing wave** is affected by changing the length, tension or mass per unit length of a vibrating string.

**Doing the experiment:**

- The equipment should be set up as shown in the diagram above.
- Start with a low mass on the mass hanger. Calculate the tension using the equation, T=mg , where T is tension in newtons, m is mass in kilograms and g is gravitational field strength in newtons per kilogram. Record the tension in a results table.
- Start the oscillator at a low frequency and gradually increase the frequency until the
**first harmonic**can be observed. Record this**frequency**in your results table. - Repeat with different masses hung on the mass hanger.
- Repeat each mass at least three times and calculate the average frequency for each tension.

**Example 1: Calculating the Frequency of the First Harmonic**

Calculate the frequency of the first harmonic of a stationary wave on a string of mass 4.0 \text{ g} and length 1.2 \text{ m} is put under 100 \text{ N} of tension.

**[3 marks]**

The equation for frequency is:

f = \dfrac{1}{2L} \times \sqrt{\dfrac{T}{\mu}}

First calculate \mu:

\mu = \dfrac{ \textcolor{f50000}{4\times 10^{-3}}}{\textcolor{f95d27}{1.2}} = \bold{3.33 \times 10^{-3}} \textbf{ kgm} \bold{^{-1}}

Substitute into the equation for frequency:

f=\dfrac{1}{2 \times \textcolor{f95d27}{1.2}} \times \sqrt{\dfrac{\textcolor{41c40e}{100}}{3.33 \times 10^{-3}}} = \bold{72.2} \textbf{ Hz}

**Example 2: Calculating the Frequency of the Third Harmonic**

Calculate the frequency of the third harmonic in a stationary wave when a string of mass 4.0 \text{ g} and length 1.2 \text{ m} is put under 100 \text{ N} of tension.

**[3 marks]**

The equation for the third harmonic is:

f=\dfrac{3}{2L} \times \sqrt{\dfrac{T}{\mu}}

First, calculate \mu:

\mu=\dfrac{\textcolor{10a6f3}{4\times 10^{-3}}}{\textcolor{aa57ff}{1.2}}=\bold{3.33\times 10^{-3}} \textbf{ kgm}\bold{^{-1}}

Substitute values into the equation:

f=\dfrac{3}{2 \times \textcolor{aa57ff}{1.2}} \times \sqrt{\dfrac{\textcolor{f21cc2}{100}}{3.33\times 10^{-3}}}=\bold{217}\textbf{ Hz}

## Stationary Waves Example Questions

**Question 1:** Explain what a stationary wave is and how it is formed.

**[3 marks]**

A stationary wave is a wave formed between **two fixed points**. It **stores energy **and forms when two identical waves meet, travelling in **opposite directions**.

**Question 2:** Describe what the 5^{th} harmonic of a stationary wave in a string would look like. You should include the number of node, anti-nodes and the wavelength.

**[3 marks]**

The 5^{th} harmonic of a stationary wave would include \bold{6}** nodes **and \bold{5}** anti-nodes**.

The wavelength would be \bold{\dfrac{2L}{5}}.

**Question 3:** Calculate the frequency of the first harmonic when the length of the string is 0.5 \text{ m}, tension is 50 \text{ N} and the mass per unit length is 0.625 \text{ kgm} ^{-1}.

**[2 marks]**

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