Solving for standing waves typically involves determining the possible wavelengths and frequencies that can exist within a given boundary or medium, such as a string fixed at both ends or an air column in a pipe. The core of this process relies on understanding the boundary conditions that define where nodes (points of no displacement) or antinodes (points of maximum displacement) must occur.
The Fundamental Relationship
For standing waves on a string fixed at both ends or sound waves in an open pipe, the length of the medium (L) is directly related to the wavelength (λ) and the harmonic number (n, an integer representing the mode of vibration). As stated in the provided reference, the relationship is:
*L = n (λ / 2)**
Here:
- L is the length of the string or pipe.
- n is the harmonic number (n = 1, 2, 3, ...). n=1 corresponds to the fundamental frequency or first harmonic, n=2 to the second harmonic, and so on.
- λ is the wavelength of the standing wave.
Solving for Wavelength
To solve for the possible wavelengths (λ) of standing waves, you can rearrange the fundamental equation:
λ = 2 * L / n
This formula, explicitly mentioned in the reference ("It's going to be 2 times l divided by n"), shows that only specific wavelengths are possible for a given length L and harmonic n.
Examples of Allowed Wavelengths:
- Fundamental Frequency (n=1): λ₁ = 2L
- Second Harmonic (n=2): λ₂ = 2L / 2 = L
- Third Harmonic (n=3): λ₃ = 2L / 3
These wavelengths correspond to different patterns of nodes and antinodes along the string or in the pipe.
Determining Frequency
Once you have solved for the possible wavelengths (λ), you can determine the corresponding frequencies (f) using the universal wave equation:
v = f * λ
where v is the speed of the wave in the medium. Rearranging to solve for frequency:
f = v / λ
By substituting the allowed wavelengths (λ = 2L/n), you can find the allowed frequencies:
f = v / (2L / n)
f = n * (v / 2L)
The term (v / 2L) represents the fundamental frequency (f₁, when n=1). Thus, the frequencies of the harmonics are integer multiples of the fundamental frequency:
fn = n * f₁
Boundary Conditions Matter
The exact method for solving standing waves depends on the specific boundary conditions:
- String fixed at both ends: Must have nodes at both ends.
- Open pipe: Must have antinodes at both ends.
- Closed pipe: Must have a node at the closed end and an antinode at the open end.
While the reference specifically focuses on the string case (L = nλ/2), the principle of relating the length to integer multiples of half-wavelengths or quarter-wavelengths based on boundary conditions is universal to solving standing wave problems.
Summary Table: String Fixed at Both Ends
| Parameter | Formula | Description |
|---|---|---|
| Wavelength | λ = 2L / n | Possible wavelengths for harmonics n = 1, 2, 3... |
| Frequency | f = n * (v / 2L) | Possible frequencies for harmonics n = 1, 2, 3... |
| Speed | v | Speed of the wave in the medium |
| Length | L | Length of the medium (string) |
| Harmonic | n | Integer (1, 2, 3...) representing the mode |
Solving for standing waves involves using the physical dimensions and constraints of the system to find the discrete set of wavelengths and frequencies that satisfy the boundary conditions, often starting with the relationship between length and wavelength as derived from the visual representation of the waves.
