**Today, we’re covering:**

- Stationary waves in strings
- Stationary Waves in tubes
- Harmonics & Overtones

*Now, we will consider stationary waves formed in columns of certain lengths.*

__We will mainly focus on 2 examples:__

Strings | |

Air columns |

Here, we will consider both examples as being a COLUMN (a length where the wave can form).

__Features of a column:__

Open end | Any wave source Will have an ANTINODE (maximum displacement) |

Closed end | Any surface/fixed point Will have a NODE (no displacement) |

__What affects the formation of stationary waves?__

Type of column | Open at 1 end OR closed at both ends |

The length of the column | Denoted by L |

Wavelength of the wave | Denoted by λ |

__Why do these factors determine whether a stationary wave will be formed?__

__TYPE OF COLUMN__

1 Open end, 1 closed end | Ex: Most close-ended air columns |

2 closed ends | Ex: Most string set-ups |

__To form a stationary wave, you need to fulfil 2 simple requirements:__

- The CLOSED end(s) of the column must coincide with a NODE of the stationary wave
- The OPEN end of the column must coincide with an ANTINODE of the stationary wave

*Thus,*The LENGTH (L) of the column must be able to accommodate these requirements in order for a stationary wave to be formed.

As you can see,

- the distance between a node & a consecutive antinode is always λ/4
- the distance between a node & a consecutive node is always λ/2

**With this, we can conclude:**

Open-ended | Close-ended |

The closed end must have a node The open end must have an antinode | Both ends must have nodes |

Length must be an ODD whole-number integer of λ/4 | Length must be ANY whole-number integer of λ/2 |

Since v = fλ, | Since v = fλ, |

Formula for f_{n}Where n = 1,2,3,.. | Formula for f_{n}Where n = 1,2,3,.. |

This way, you can find the frequencies where a stationary wave can form, given a way to get the values of L, v & n.

__What is the lowest allowed frequency for a stationary wave called?__

The FUNDAMENTAL FREQUENCY.

This is when n = 1.

__What are the allowed frequencies for stationary waves called?__

HARMONICS or OVERTONES.

There is a difference in usage:

Harmonics | All the allowed frequencies. The nth harmonic = f _{n}So, the first harmonic is f _{1 }or the fundamental frequency. |

Overtones | The allowed frequencies ABOVE the fundamental frequency. The nth overtone = f _{n+1}So, the first overtone is f _{2 }or the first allowed frequency above the fundamental frequency. |

__Solving problems involving harmonics:__

There is a simple way to deal with these problems!

STEP 1: Identify the type of column | Is it open at one end or closed at both ends? |

STEP 2: Write out the correct formula depending on the type of column | For open-ended, use For close-ended, use |

STEP 3: List out the known values, & sub into the appropriate places in the formula | At least 2 variables of the 4 must be given, in order to set up a simultaneous equation of 2 unknowns. For example, you may be given consecutive frequencies along with the length of the column. Ex:given consecutive resonant frequencies 135Hz & 180Hz in a close-ended string of length 1.20m: |

STEP 4: Set up a simultaneous equation, & find the required values | Notice which values remain constant for a given question. Usually the velocity of the wave v does not change (if it remains in the same medium), as well as the length L. Therefore, Since k is the same for both: You can solve for n! |

Next, we’ll see how to measure the speed of sound experimentally using stationary waves.

**⇐ Previous in Physics: Stationary Waves (Intro)****⇒ Next in Physics: Measuring the speed of sound using stationary waves**

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