# Worksheet: Normal Vibrational Modes

In this worksheet, we will practice calculating the modal numbers and the positions of nodes and antinodes in standing waves.

**Q1: **

An organ pipe of length 3.00 m is closed at both ends.

The speed of sound is 343 m/s.

Compute the wavelength of the first mode of resonance.

Compute the wavelength of the second mode of resonance.

Compute the wavelength of the third mode of resonance.

Compute the frequency of the first mode of resonance.

Compute the frequency of the second mode of resonance.

Compute the frequency of the third mode of resonance.

**Q4: **

A nylon guitar string is fixed between two lab posts 2.00 m apart. The string has a linear mass density of 7.20 g/m and is placed under a tension of 160.0 N. The string resonates at the mode when it is plucked. The string is placed next to a tube of length that is open at both ends, and the sound wave produced by the vibrating string resonates with the fundamental mode of the tube. Find , using a value of 343 m/s for the speed of sound in the tube.

**Q6: **

A string with a length of 4.0 m is held under a constant tension. The string has a linear mass density of 0.0060 kg/m. Two resonant frequencies of the string are 400 Hz and 480 Hz. There are no resonant frequencies between the two frequencies.

What is the wavelength of 400 Hz mode?

What is the wavelength of the 480 Hz mode?

What is the tension in the string?

**Q7: **

A string on the violin has a length of 23.0 cm and a mass of 0.900 grams. The tension in the string is 850 N. The string is plucked and oscillates in the mode. The temperature in the room is .

What is the speed of the wave on the string?

What is the wavelength of the standing wave produced on the string?

What is the frequency of the oscillating string?

What is the frequency of the sound produced?

What is the wavelength of the sound produced?

**Q8: **

The air column in a tube, which is closed at one end, has a fundamental frequency of 256 Hz. Some students fill the tube with water and then slowly lower the water level by draining the tube. Whilst lowering the water level, the students repeatedly ring a tuning fork of frequency 256 Hz and listen for resonance with the air column. When the air column reaches a length of 0.336 m, resonance first occurs.

What is the air temperature in the tube?

If the water level continues to lower, what length of air column will produce the second occurrence of resonance?

**Q9: **

A tube that is 4.0 m long and open at one end and closed at the other is in a room where the temperature is . A speaker capable of producing variable frequencies is placed at the open end and is used to cause the tube to resonate.

What is the wavelength of the fundamental frequency of the tube?

What is the fundamental frequency of the tube?

What is the wavelength of the first overtone of the tube?

What is the frequency of the first overtone of the tube?

**Q12: **

As a crude approximation, voice production can be modeled as sound waves resonating in a tube that is closed at one end. A tube length of 0.150 m models the typical length of the breathing passages and mouth used to generate speech. For modeling voice production, use a value of 8.314 J/mol⋅K for the universal gas constant and for the temperature in the tube.

Determine the fundamental frequency of the voice production tube for a 0.150 m tube, taking the air temperature to be .

Determine the fundamental frequency of the voice production tube if helium is contained in the tube. Use a value 1.40 for the specific heat ratio of helium and 4.00 g/mol for its molar mass. Assume the same temperature dependence of wave speed in helium as in air.

**Q14: **

The ear canal resonates like a tube closed at one end. Ear canals range in length from 1.62 to 2.43 cm in an average population.

Determine the lower bound of the range of fundamental resonant frequencies for an air temperature of .

Determine the upper bound of the range of fundamental resonant frequencies for an air temperature of .

**Q15: **

A tube is capped with a movable piston, creating a tube with a variable length. A tuning fork of frequency 256 Hz is struck and placed next to the tube and the piston is slid down the tube until resonance occurs. Resonance occurs when the piston is 231 cm from the tube’s open end. The piston is slid further down the tube until the resonance occurs again; this time, the piston is 165 cm from the open end of the tube.

What is the speed of sound in the tube?

How far from the open end of the tube would the piston be when resonance next occurs?

**Q16: **

A speaker powered by a signal generator is used to study resonance in a tube. The signal generator can be adjusted from a frequency of 1,024 Hz to 2,000 Hz. The speaker first vibrates air in a 0.95-meter tube that is open at both ends, and then one end of the tube is closed. The temperature in the room is .

What is the smallest normal mode number of the pipe that can be produced with both ends of the tube open?

What is the largest normal mode number of the pipe that can be produced with both ends of the tube open?

What is the largest normal mode number of the pipe that can be produced with one end of the tube closed?

**Q17: **

A string of length 2.00 m is stretched between two supports. The string has a tension that produces waves that have a speed of 50.0 m/s.

What is the wavelength of the first resonant mode on the string?

What is the wavelength of the second resonant mode on the string?

What is the wavelength of the third resonant mode on the string?

What is the frequency of the first resonant mode on the string?

What is the frequency of the second resonant mode on the string?

What is the frequency of the third resonant mode on the string?

**Q20: **

Consider two wave functions and , where is measured in meters and is measured in seconds. The resultant wave form of the two functions is . In the case of , , , and .

The positions of the nodes of the wave function are measured from the point , in the positive -direction.

What is the position of the first node of the resultant wave form?

What is the position of the second node of the resultant wave form?

What is the position of the third node of the resultant wave form?

**Q23: **

A 0.800 m long tube is opened at both ends. The air temperature is . The air in the tube is oscillated using a speaker attached to a signal generator.

Find the wavelength of the first resonant mode of the tube.

Find the wavelength of the second resonant mode of the tube.

Find the frequency of the first resonant mode of the tube.

Find the frequency of the second resonant mode of the tube.

**Q24: **

A string of mass 0.00900 kg and length 3.00 m is fixed at both ends. It is under a tension of 200.00 N and is driven by a variable frequency source to produce standing waves on the string.

Find the wavelength of the first mode of standing waves on the string.

Find the wavelength of the second mode of standing waves on the string.

Find the wavelength of the third mode of standing waves on the string.

Find the wavelength of the fourth mode of standing waves on the string.

Find the frequency of the first mode of standing waves on the string.

Find the frequency of the second mode of standing waves on the string.

Find the frequency of the third mode of standing waves on the string.

Find the frequency of the fourth mode of standing waves on the string.

**Q25: **

A 2.5-meter pipe, open at both ends, is placed in a room at a temperature . A speaker that is capable of producing variable frequencies is placed at the open end and is used to cause the tube to resonate.

What is the fundamental frequency of the pipe?

What is the wavelength of the first overtone of the pipe?