# Stationary Waves in Air Columns A **stationary wave** (or standing wave) is formed by the superposition of two identical waves traveling in opposite directions. This interference pattern results in points of zero displacement called **nodes** and points of maximum displacement called **antinodes**. - **Transverse Stationary Waves**: Can be formed on a string fixed at both ends. The particles vibrate perpendicular to the string's length. - **Longitudinal Stationary Waves**: Formed in air columns (like organ pipes). Air particles oscillate parallel to the length of the pipe, creating nodes (points of no displacement/high pressure variation) and antinodes (points of maximum displacement/low pressure variation). ## Closed-End Organ Pipe In an organ pipe that is closed at one end and open at the other: - A **displacement node** is always formed at the **closed end**. - A **displacement antinode** is always formed at the **open end**. Because of these boundary conditions, a closed-end pipe can support only **odd harmonics** (1st, 3rd, 5th, ...). ### 1st Harmonic (Fundamental Frequency) - This is the simplest standing wave pattern, with one node and one antinode. - The length of the pipe ($L$) is equal to one-quarter of a wavelength ($\lambda_1$): $$L = \frac{\lambda_1}{4} \quad \implies \quad \lambda_1 = 4L$$ - Using the wave speed equation $v = f_1 \lambda_1$, the fundamental frequency is: $$f_1 = \frac{v}{4L}$$ ### 3rd Harmonic (First Overtone) - The next possible standing wave has an additional node and antinode. - The length of the pipe contains three-quarters of a wavelength ($\lambda_3$): $$L = \frac{3\lambda_3}{4} \quad \implies \quad \lambda_3 = \frac{4L}{3}$$ - The frequency for this harmonic is: $$f_3 = \frac{v}{\lambda_3} = \frac{v}{4L/3} = \frac{3v}{4L}$$ - This frequency is three times the fundamental frequency: $$f_3 = 3f_1$$ ### 5th Harmonic (Second Overtone) - The next pattern contains five-quarters of a wavelength ($\lambda_5$): $$L = \frac{5\lambda_5}{4} \quad \implies \quad \lambda_5 = \frac{4L}{5}$$ - The frequency for the 5th harmonic is: $$f_5 = \frac{v}{\lambda_5} = \frac{v}{4L/5} = \frac{5v}{4L}$$ - This frequency is five times the fundamental frequency: $$f_5 = 5f_1$$ ## Open-End Organ Pipe In an organ pipe that is open at both ends: - A **displacement antinode** is formed at **both open ends**. This configuration allows an open-end pipe to support **all harmonics** (both odd and even). ### 1st Harmonic (Fundamental Frequency) - The simplest pattern has an antinode at each end and one node in the middle. - The length of the pipe is equal to half a wavelength ($\lambda_1$): $$L = \frac{\lambda_1}{2} \quad \implies \quad \lambda_1 = 2L$$ - The fundamental frequency is: $$f_1 = \frac{v}{2L}$$ ### 2nd Harmonic (First Overtone) - The next standing wave has two nodes. - The length of the pipe is equal to one full wavelength ($\lambda_2$): $$L = \lambda_2$$ - The frequency for the 2nd harmonic is: $$f_2 = \frac{v}{L} = \frac{v}{2L} \times 2$$ - This frequency is twice the fundamental frequency: $$f_2 = 2f_1$$ ### 3rd Harmonic (Second Overtone) - This pattern has three nodes. - The length of the pipe is equal to three-halves of a wavelength ($\lambda_3$): $$L = \frac{3\lambda_3}{2} \quad \implies \quad \lambda_3 = \frac{2L}{3}$$ - The frequency for the 3rd harmonic is: $$f_3 = \frac{v}{\lambda_3} = \frac{v}{2L/3} = \frac{3v}{2L}$$ - This frequency is three times the fundamental frequency: $$f_3 = 3f_1$$ ## End Correction In practical scenarios, the antinode does not form exactly at the open end but slightly outside it. This is known as **End Correction**. ## Summary Table | Pipe Type | Boundary Conditions | Harmonics Present | Fundamental Frequency ($f_1$) | Nth Harmonic Frequency ($f_n$) | | :--- | :--- | :--- | :--- | :--- | | **Closed-End** | Node at closed end, Antinode at open end | Odd only (1, 3, 5, ...) | $f_1 = \frac{v}{4L}$ | $f_n = n f_1$ (for odd n) | | **Open-End** | Antinode at both ends | All (1, 2, 3, ...) | $f_1 = \frac{v}{2L}$ | $f_n = n f_1$ (for all n) |