Standing Waves Practice Problems

 A physics student plucks a taut 1-meter string fixed at both ends and observes the oscillations. The student measured its mass as 4 g and length as 1 m. The fundamental frequency and amplitude are measured using a high-speed camera coupled to an image processor. The fundamental frequency produced by the oscillating string is 80 Hz, and the amplitude at an antinode is 6 cm. Determine the speed of the waves in the string.

A 0.8 m string with a mass of 1.6 g is attached to a tuning peg on one end and a wooden board on the other. The peg is turned clockwise to obtain an 80-N tension in the string. Calculate the frequency of the fundamental mode of oscillation.

A copper cable is tied between two trees. The horizontal distance between the two trees is 3 m. The transverse waves travel at a speed of 80 m/s along the taut cable. Calculate the i) wavelength and ii) frequency of the third overtone.

A violin's nylon string of length 0.51 m is fixed at both ends. A transverse wave on this string travels at a speed of 600 m/s. Determine i) the wavelength and ii) the frequency of the third harmonic.

The speed of transverse waves in a stretched guitar string is 417 m/s. Determine the first harmonic i) wavelength and ii) frequency if the string length is 75 cm.

A physicist uses a 40 cm plastic line with a linear density of 0.3 g/m to create sounds by making the line vibrate. The physicist tests the instrument in a lecture hall by making the line vibrate in its fundamental mode. The temperature in the hall was 20°C. The students in the hall catch a sound with a wavelength of 50 cm. Calculate the tension in the line.

A wire of length 0.8 m and mass 10 g is connected at one end to a spring and at the other end to a clamp. The wire is stretched horizontally to elongate the spring by 6.0 cm. When it is strummed, the wire vibrates in its fundamental mode at a frequency of 120 Hz. Calculate the spring constant. Assume that the spring is at rest.

When a plastic line of length L is stretched to a tension of T between two clamps, the frequency of the first harmonic is 192 Hz. The tension is tripled by turning the clamps. Calculate the new frequency of the first harmonic.

Opposing waves with equal frequency and amplitude form a standing wave via interference. The figure below shows a standing wave vibrating on a string with an oscillation frequency of 250 Hz. Calculate the speed at which this wave travels along the string.

Antinodes are points of maximum displacement in standing waves. In the figure below, a standing wave on a string oscillates at a frequency of 200 Hz. If the frequency were to be doubled to 400 Hz, keeping the wave's speed constant how many antinodes would be present in the resulting wave pattern?

Consider a rope that is anchored at both ends, with a total length of 72 cm. Using the concept of standing waves, calculate the four longest wavelengths that can be produced on this rope.

For a 2.2-meter rope secured at both ends, standing waves are observed at two consecutive frequencies,30 Hz and 45 Hz. Draw an illustration of the standing-wave pattern that emerges when the rope resonates at the higher frequency of 45 Hz.

A guitarist is tuning his acoustic guitar and pays particular attention to two strings: the 5th string (A string), which is tuned to 110 Hz, and the 2nd string (B string), tuned to 246 Hz. Considering that the strings have the same length and are subject to nearly equal tension, estimate the mass ratio between the A string and the B string.

A research lab is investigating the properties of an erbium-doped fiber laser (EDFL) for potential applications in telecommunications. The lab has set up an EDFL with a cavity length of 48 cm, configured to oscillate in the 150,000 modes. To better understand the performance of this laser, the researchers need to determine the wavelength and frequency of the laser beam generated by the EDFL. Calculate these parameters for the given erbium-doped fiber laser setup.

Consider an inextensible and massless string of length L suspended from the top of a roof. A block of mass m_b is firmly attached to the string's lower end. When the string is wiggled, a third-harmonic standing wave of frequency 250 Hz is set up on the string. If an additional mass of 1.5 kg is added to the lower end, the third harmonic frequency is shifted to 285 Hz. Calculate the block's mass.

Consider a guitar string made from an elastic material that can support standing waves when stretched between two fixed points. This guitar string is 14 cm long and has a cross-sectional area of 78 mm². The material's density is 980 kg/m³. When the string is under a tension of 400 N, find the lowest frequency of the guitar string.

A cello string has a fundamental frequency of 420 Hz when played open. If the string is fingered one-third of the way down from the end, only two-thirds of the string length vibrates. What is the new fundamental frequency of the string under these conditions?

What is the distance between two consecutive nodes on a guitar string where a wave travels at a speed of 98 m/s with a standing wave frequency of 452 Hz?

What are the frequencies of the fundamental and the first two overtones for a violin string that is 80 cm long and has a mass of 3.16 g with a vibrating portion of 60 cm under a tension of 510 N?

Alex experimented with waves on a long rope in his physics lab. He observed the formation of standing waves and noted the displacement of the wave as: D=2.8 sin (0.70x) cos (44t) where t is in seconds and x and D are in centimeters. Determine the distance between two adjacent nodes?

Jordan experimented with waves on a long string in his physics lab. He observed the formation of standing waves and noted the displacement of the wave as: D=3.0 sin(0.70x) cos(46t) where t is in seconds and x and D are in centimeters. Provide the amplitude, frequency, and speed of each component wave.

What will be the new frequency of a banjo string that initially plays at 310 Hz if the tension is increased by 15%?

When a 2.25-meter-long horizontal string vibrates at 200 Hz, it forms a standing wave exhibiting five loops. Swinging from top to bottom, a maximum displacement of 12 cm is achieved at the center of each loop. Determine the function representing this standing wave.

A standing wave is created in a 60-cm-wide tub with a frequency of 0.75 Hz. The wave oscillates up and down at both ends of the tub with minimal movement at the center of the tub. What is the speed of this wave?

The intended frequency for a violin string is 600 Hz, but during play, it vibrates at 660 Hz. What percentage adjustment to the tension is needed to achieve the correct frequency?

Identify at least three other frequencies at which a piano string vibrating in four segments at a frequency of 300 Hz will resonate, and specify the name of each frequency.

Given two piano strings, one set to produce a C note at 261 Hz and the other an F note at 349 Hz, what are the frequencies of their respective first two overtones?

Determine the ratio of the lengths of violin strings (L_E / L_A), if The E string of a violin vibrates at 660 Hz while the A string vibrates at 880 Hz. Assume both strings have the same mass per unit length and experience similar tension,

A horizontal rope is attached to a 70.0-Hz mechanical shaker with a small amplitude. The rope has a mass per unit length of $3.0\times 10^{-4}\ \frac{kg}{m}$. It extends over a fixed support located 1.75 m from the shaker, and a mass is attached to the rope's free end, as illustrated below. What mass m should be suspended from this end to form 7 loops of a standing wave? Assume the end at the shaker is a node.

Determine the formula for the wave pulse at any time t of a transverse wave traveling to the right along a rope at 2.0 meters per second. At time t = 0, the pulse is defined by $D=\frac{6.0\ m^3}{(x^2+4.0\ m^2)}$, where D and x are in meters. Assume no frictional losses.

The displacement of a ripple traveling through a shallow pool is described by:

$D_1=5.1\sin (0.96y-25t+3.2)$, where $D_1$ and $y$ are in centimeters and $t$ is in seconds. Determine an equation for a ripple moving in the opposite direction that will form a standing wave when combined with this one.

A student builds a tunable stringed musical instrument using a fishing line of mass 6 g and length 60 cm. The fishing line slides through a tuning peg that allows the student to adjust the tension. The developed instrument was tested in a room where the speed of sound is 344 m/s. The student adjusts the tension so that when it vibrates in its second overtone, it produces sound with a wavelength of 0.63 m. i) Determine the tension in the fishing line in order to vibrate in the second overtone. ii) Determine the frequency of the sound produced by this line in its fundamental mode of vibration.

A guitarist is tuning his guitar using harmonics. When in tune, the sixth string (low E string) should be at a frequency of 82.41 Hz, and the fourth string (D string) should be at a frequency of 146.83 Hz. Determine the frequency difference between the fourth harmonic of the sixth string and the third harmonic of the fourth string.

Determine the speed of the traveling waves on a 75-cm banjo string that is fixed at both ends and resonates at frequencies of 2.2, 2.5, and 2.8 kHz within the range of 2.0 to 3.0 kHz.