Table of contents  
Reflection of Waves  
Standing Waves & Normal Modes  
Stretched String fixed at Both Ends  
Air Column with One End Closed 
Reflection of Waves
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.
102 videos411 docs121 tests

1. What is the reflection of waves? 
2. What are standing waves and normal modes? 
3. How is a stretched string fixed at both ends related to standing waves? 
4. How does an air column with one end closed create standing waves? 
5. How do reflection of waves and standing waves play a role in music and acoustics? 

Explore Courses for NEET exam
