Problem 30 - University Physics with Modern Physics

Question

singing in the Shower. A pipe closed at both ends can have standing waves inside of it, but you normally don't hear them because little of the sound can get out. But you can hear them if you are inside the pipe, such as someone singing in the shower. (a) Show that the wavelengths of standing waves in a pipe of length \(L\) that is closed at both ends are \(\lambda_{n}=2 L / n\) and the frequencies are given by \(f_{n}=n v / 2 L=n f_{1},\) where \(n=1,2,3, \ldots,\) (b) Modeling it as a pipe, find the frequency of the fundamental and the first two overtones for a shower 2.50 \(\mathrm{m}\) tall. Are these frequencies audible?

Step-by-Step Solution

Answer

The fundamental frequency is 68.6 Hz, and the first two overtones are 137.2 Hz and 205.8 Hz; all are audible.

Step 1: Understanding the Problem

We need to find the wavelengths and frequencies of standing waves in a pipe closed at both ends. The problem states we need to derive the relationship for wavelengths \( \lambda_n \) and frequencies \( f_n \). Afterward, for a pipe of length 2.50 m, we find the frequencies of the fundamental and first two overtones.

Step 2: Deriving the Wavelengths Formula

For a pipe closed at both ends, a standing wave has nodes at each end. The possible wavelengths are determined by the requirement that the length \( L \) of the pipe is an integer multiple of half-wavelengths: \( L = n \frac{\lambda_n}{2} \), where \( n \) is a positive integer. Solving for \( \lambda_n \), we find \( \lambda_n = \frac{2L}{n} \).

Step 3: Deriving the Frequencies Formula

The frequency \( f_n \) of a wave is related to its wavelength \( \lambda_n \) and the speed of sound \( v \) by the equation \( f_n = \frac{v}{\lambda_n} \). Substituting \( \lambda_n = \frac{2L}{n} \), we get \( f_n = \frac{nv}{2L} \). This expression is also \( f_n = nf_1 \), where \( f_1 = \frac{v}{2L} \) is the fundamental frequency.

Step 4: Calculating the Fundamental Frequency

Given that the length \( L = 2.50 \) m and taking the speed of sound \( v \approx 343 \) m/s (at room temperature), the fundamental frequency \( f_1 \) is \( f_1 = \frac{343}{2 \times 2.50} = 68.6 \) Hz.

Step 5: Calculating the Frequencies of the First Two Overtones

The first overtone frequency is \( f_2 = 2 \times f_1 = 137.2 \) Hz, and the second overtone frequency is \( f_3 = 3 \times f_1 = 205.8 \) Hz.

Step 6: Determining Audibility

The human ear can generally hear frequencies between 20 Hz and 20,000 Hz. Thus, the fundamental frequency and the first two overtones (68.6 Hz, 137.2 Hz, 205.8 Hz) are all within the audible range.

Key Concepts

Harmonics

Harmonics are the different modes of vibration that strings or air columns can have. When we talk about harmonics in the context of standing waves in pipes, we're referring to the specific frequencies at which these vibrations occur. Every harmonic is a multiple of the fundamental frequency, which means each harmonic corresponds to a different standing wave pattern. For a pipe closed at both ends, these harmonics appear as integer multiples of a fundamental vibration mode. This integration of various harmonics gives richness to the sound we hear. In essence, harmonics shape the unique quality or timbre of a sound.

Fundamental Frequency

The fundamental frequency, often denoted as \( f_1 \), is the lowest frequency at which a system vibrates. In the closed pipe scenario, it is determined by the speed of sound inside the pipe and its length.

• The formula for calculating this is given by \( f_1 = \frac{v}{2L} \).
• Here, \( v \) represents the speed of sound, and \( L \) is the length of the pipe.

The fundamental frequency sets the stage for all higher harmonics. Any wave that vibrates at this frequency forms the simplest standing wave pattern. It is this base frequency upon which all other overtones or harmonics are built. In practical terms, this means the sound you hear from the pipe has its root in this fundamental frequency.

Overtones

Overtones are frequencies higher than the fundamental frequency and are integral multiples of it. In the series of frequencies produced within the pipe, overtone frequencies can be represented as:

• First Overtone: \( f_2 = 2f_1 \)
• Second Overtone: \( f_3 = 3f_1 \)

These overtones contribute to the complexity and fullness of sound. The overtones in any sound add depth and character to what you hear; they are why different musical instruments have different sounds even if they play the same note.

Speed of Sound

The speed of sound is crucial in determining how waves move in a medium, like air or water. It represents the speed at which sound waves travel through a medium.

• At room temperature and standard atmospheric pressure, the speed of sound in air is approximately 343 meters per second.
• This speed can vary depending on factors such as temperature, atmospheric pressure, and the medium itself.

This constant is used to calculate the frequencies of standing waves in a pipe because both depend on how quickly vibrations can travel through the air inside the pipe. Knowing the speed of sound allows us to easily determine the frequencies associated with different lengths of pipes, making it an essential factor in acoustical calculations.

Acoustics

Acoustics is the branch of physics concerned with the study of sound. When we talk about standing waves in a pipe closed at both ends, we're dealing with a specific acoustical scenario: one where sound waves are reflected at the ends of the pipe to create standing waves.

• This involves understanding how sound waves interfere and create patterns within confined spaces.
• These patterns lead to the phenomenon of resonance, which occurs when the sound within the pipe vibrates at certain frequencies.

Such resonance is why certain frequencies become prominent and audible even when you're inside structures like a shower. By studying acoustics, we gain insights into how to manipulate sound environments, ideal for creating better musical instruments and enhancing architectural designs that rely on sound properties.