Table of contents  
Standing Waves  
Stretched String fixed at Both Ends  
Air Column with One End Closed  
Beats 
In certain situations, like a string fixed at both ends or an air column in a closed pipe, waves can bounce back and forth multiple times, creating what we call standing waves or stationary waves. Imagine a wave traveling in one direction along a string, reflecting at one end, then bouncing back from the other end. This process repeats until a steady wave pattern forms on the string.
Stationary Waves Arising from Superposition
Nodes
The positions of nodes (where the amplitude is zero) in a standing wave are given by the condition sin(kx) = 0 , implying kx = nπ where ( n = 0, 1, 2, 3, .....).
Since k = 2π/λ, we get x = nλ/2 , where n = 0, 1, 2, 3, .....
The distance between any two consecutive nodes is λ/2.
Antinodes
Similarly, the positions of antinodes (where the amplitude is largest) are given by the condition
sin(kx)= 1 , implying kx = (n + 1/2)π where ( n = 0, 1, 2, 3, .....).
With k = 2π/λ, we get x = (n+1/2)λ/2, where ( n = 0, 1, 2, 3, .....).
The distance between any two consecutive antinodes is λ/2.
For a stretched string fixed at both ends, Taking one end to be at x = 0, the boundary conditions are that x = 0 and x = L are positions of nodes. The x = 0 condition is already satisfied.
The First six harmonics of a Stretched String Fixed at Both Ends
Musical instruments, like sitars or violins, use these principles, producing a superposition of different modes when plucked or bowed. The illustration in Figure 15.13 depicts the first six harmonics of a stretched string fixed at both ends.
Example 1: The standing wave pattern is observed in a stretched string fixed at both ends, as shown in the figure below. If the speed of the waves in the string is 2m/s, at what frequency is this standing wave vibrating?
Answer: In the standing wave pattern shown, there are four loops or antinodes. These loops fit into the length of the string L. On the other side, recall that each loop has a length of ℓ=. So, the whole length of this string can accommodate four of these loops i.e. L=4ℓ or L=4(λ/2).
The wave speed and wavelength are also related together by λ=v/f. Substituting this into the length formula above, we get
Rearranging this relation and putting the numerical values, the required frequency is obtained as
Example 2: Two successive overtones of a vibrating guitar string are 280Hz and 350Hz. Find the frequency of the first harmonic?
Solution: Overtone is a name for other frequencies except the fundamental frequency. Suppose f_{n}=280Hz, where n is not equal to 1, to be an unknown overtone. The next overtone after this is obtained by increasing n by one as n>n+1. So, the other next overtone is f_{n+1}=350Hz.
Standing wave formula relates the other overtones to the first harmonic or fundamental frequency as f_{n}=nf_{1}. Divide those two overtones by each other and solve for n.
Solving the above equation, gets
This indicates that this particular standing wave formed on the string has four antinode or four loops.
Using the value of one of the given above overtone and substitute n=4 into it, we can find the fundamental frequency, f_{1}.
Hence, the frequency of the first harmonic is 70Hz.
Let's now consider the normal modes of oscillation for an air column with one end closed and the other open, such as a partially filled glass tube. In this system, the end in contact with water serves as a node (maximum pressure change, zero displacement), while the open end acts as an antinode (least pressure change, maximum displacement).
Normal Modes of an Air Column Open at One End and Closed at Other
For a pipe open at both ends, both ends act as antinodes, producing all harmonics.
Standing Waves in an Open Pipe
Both string and air column systems can also experience forced oscillations. When an external frequency is close to one of the natural frequencies, the system exhibits resonance.
In the case of a circular membrane, like a tabla, the normal modes are determined by the condition that no point on the membrane's circumference vibrates. Estimating the frequencies of these modes is more complex, involving wave propagation in two dimensions, although the underlying physics remains the same.
'Beats' is an interesting thing that happens when two sound waves with similar but not exactly equal frequencies overlap.
Superposition of 11Hz and 9Hz Harmonic Waves
Example 3: Find the beat frequency when two waves have frequencies of 550Hz and 380Hz.
Answer: With frequencies f_{1} = 550Hz and f_{2} = 380Hz,
The beat frequency (f_{b}) is calculated as the absolute difference between the two frequencies: 380  550 = 170Hz.
Therefore, the beat frequency is 170Hz.
102 videos411 docs121 tests

1. What are standing waves and how are they formed in a stretched string fixed at both ends? 
2. How are standing waves produced in an air column with one end closed? 
3. What are normal modes in the context of standing waves and how are they related to the frequency of the waves? 
4. How do beats occur in the context of standing waves and how are they used in music tuning? 
5. How can the concept of standing waves be applied in reallife scenarios outside of academic contexts? 

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