PHY C15: Stationary Waves in Strings/Air Columns

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 Image result for stationary waves reflect gif
Air columns Image result for stationary waves reflect gif
By Dan Russell

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

Features of a column:

Open endAny wave source
Will have an ANTINODE (maximum displacement)
Closed endAny surface/fixed point
Will have a NODE (no displacement)

What affects the formation of stationary waves?

Type of columnOpen at 1 end OR closed at both ends
The length of the columnDenoted by L
Wavelength of the waveDenoted 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
Image result for stationary waves reflect gif
2 closed ends  Ex: Most string set-ups
Image result for stationary waves reflect gif

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.

From KhanAcademy

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-endedClose-ended
This image has an empty alt attribute; its file name is screenshot-2020-02-10-11.16.31.png  This image has an empty alt attribute; its file name is e45c33bb944a2f5fd7a853d18fd6f15d05cd0adf.png
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
undefined
Since v = fλ,
undefined
Since v = fλ,
undefined
Formula for fnundefined
Where n = 1,2,3,..
Formula for fn

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:

HarmonicsAll the allowed frequencies.

The nth harmonic = fn

So, the first harmonic is f1 or the fundamental frequency.
OvertonesThe allowed frequencies ABOVE the fundamental frequency.

The nth overtone = fn+1

So, the first overtone is f2 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
undefined

For close-ended, use
undefined
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:
undefined
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,
undefined
Since k is the same for both:
undefined
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

3 thoughts on “PHY C15: Stationary Waves in Strings/Air Columns

  1. Pingback: PHY C15: Measuring the Speed of Sound using Stationary Waves – ProDuckThieves

  2. Pingback: 15. Superposition – ProDuckThieves

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