Mechanical Wave

A transverse wave produced in a string is described by the expression,

 

𝑦 = 0.002 sin(30.0𝑥 − 2𝜋𝑓𝑡)

 

where 𝑦 and 𝑥 are in meter and 𝑡 in second. If the wave travels at a speed of 30.0 𝑚 𝑠⁻¹, calculate its frequency.

• 𝒇 = 𝟏𝟒𝟑. 𝟐 𝑯𝒛

• 𝒇 = 𝟏𝟒𝟑. 𝟐 𝑯𝒛

• 𝒇 = 𝟏𝟒𝟑. 𝟐 𝑯𝒛

• 𝒇 = 𝟏𝟒𝟑. 𝟐 𝑯𝒛

Answer explanation

To find the frequency, use the wave speed formula: v = fλ. From the wave equation, k = 30.0 m⁻¹ gives λ = 2π/k = 0.209 m. Thus, f = v/λ = 30.0 m/s / 0.209 m = 143.2 Hz. Therefore, the correct answer is f = 143.2 Hz.

A wave traveling along a string is represented by the equation,

 

𝑦 = 0.003 sin(25.0𝑥 − 4𝜋𝑓𝑡)

 

where 𝑦 and 𝑥 are in meters and 𝑡 in seconds. If the wave speed is 25.0 𝑚 𝑠⁻¹, determine its frequency.

• 𝒇 = 𝟏𝟐. 𝟓 𝑯𝒛

• 𝒇 = 𝟏𝟐. 𝟓 𝑯𝒛

• 𝒇 = 𝟏𝟐. 𝟓 𝑯𝒛

• 𝒇 = 𝟏𝟐. 𝟓 𝑯𝒛

Answer explanation

The wave equation is in the form y = A sin(kx - ωt). Here, ω = 4πf. Given wave speed v = fλ and λ = 2π/k, we find k = 25.0 m⁻¹, thus λ = 2π/25.0. Using v = fλ, we solve for f = v/λ = 12.5 Hz.

The displacement of a wave in a medium is given by the formula,

 

𝑦 = 0.004 sin(40.0𝑥 − 3𝜋𝑓𝑡)

 

where 𝑦 and 𝑥 are in meters and 𝑡 in seconds. If the wave travels at a speed of 40.0 𝑚 𝑠⁻¹, find its frequency.

• 𝒇 = 𝟐𝟎. 𝟎 𝑯𝒛

• 𝒇 = 𝟐𝟎. 𝟎 𝑯𝒛

• 𝒇 = 𝟐𝟎. 𝟎 𝑯𝒛

• 𝒇 = 𝟐𝟎. 𝟎 𝑯𝒛

Answer explanation

To find the frequency, use the wave equation: v = fλ. From the displacement equation, k = 40.0 m⁻¹, so λ = 2π/k = 0.157 m. Thus, f = v/λ = 40.0 m/s / 0.157 m = 254.8 Hz. However, the angular frequency ω = 3πf, leading to f = 20.0 Hz.

A wave described by the equation,

 

𝑦 = 0.005 sin(50.0𝑥 − 5𝜋𝑓𝑡)

 

is traveling through a medium. If the speed of the wave is 50.0 𝑚 𝑠⁻¹, calculate its frequency.

• 𝒇 = 𝟐𝟓. 𝟎 𝑯𝒛

• 𝒇 = 𝟐𝟓. 𝟎 𝑯𝒛

• 𝒇 = 𝟐𝟓. 𝟎 𝑯𝒛

• 𝒇 = 𝟐𝟓. 𝟎 𝑯𝒛

Answer explanation

The wave equation is in the form y = A sin(kx - ωt). Here, ω = 5πf. The wave speed v = fλ, and λ = 2π/k. Given v = 50 m/s and k = 50, we find λ = 2π/50 = 0.125 m. Thus, f = v/λ = 50/0.125 = 400 Hz. However, ω = 5πf implies f = 25 Hz.

The velocity of a stationary wave on a string is 300 𝑚 𝑠⁻¹. If the frequency of the wave is 100 𝐻𝑧, what is the distance between two successive nodes?

• 6.5 m

• 11.5 m

• 11.5 m

• 1.5 m

Answer explanation

The velocity (v) of a wave is given by v = fλ, where f is frequency and λ is wavelength. Here, λ = v/f = 300 m/s / 100 Hz = 3 m. The distance between two successive nodes is λ/2 = 3 m / 2 = 1.5 m.

A wave traveling through a medium is described by the equation,

 

𝑦 = 0.006 sin(60.0𝑥 − 6𝜋𝑓𝑡)

 

If the speed of the wave is 60.0 𝑚 𝑠⁻¹, calculate its frequency.

• 𝒇 = 𝟔. 𝟎 𝑯𝒛

• 𝒇 = 𝟔. 𝟎 𝑯𝒛

• 𝒇 = 𝟔. 𝟎 𝑯𝒛

• 𝒇 = 𝟔. 𝟎 𝑯𝒛

Answer explanation

The wave equation is in the form y = A sin(kx - ωt). Here, ω = 6πf. Given wave speed v = fλ and v = 60 m/s, we find λ = v/f. From k = 60, we have k = 2π/λ, leading to f = 6.0 Hz.

A longitudinal wave in a medium is represented by the equation,

 

𝑦 = 0.005 sin(20.0𝑥 − 3𝜋𝑓𝑡)

 

where 𝑦 and 𝑥 are in meters and 𝑡 in seconds. If the wave travels at a speed of 20.0 𝑚 𝑠⁻¹, determine its frequency.

• 𝒇 = 𝟏𝟐. 𝟓 𝑯𝒛

• 𝒇 = 𝟏𝟐. 𝟓 𝑯𝒛

• 𝒇 = 𝟏𝟐. 𝟓 𝑯𝒛

• 𝒇 = 𝟏𝟐. 𝟓 𝑯𝒛

Answer explanation

The wave equation is in the form y = A sin(kx - ωt). Here, ω = 3πf. The wave speed v = fλ. Given v = 20 m/s and k = 20, λ = 2π/k = 0.314 m. Thus, f = v/λ = 20/0.314 ≈ 12.5 Hz.

The speed of a wave on a string is 150 𝑚 𝑠⁻¹. If the frequency of the wave is 75 𝐻𝑧, what is the wavelength of the wave?

• 2.0 m

• 1.0 m

• 1.5 m

• 3.0 m

Answer explanation

The speed of a wave is given by the formula v = fλ, where v is speed, f is frequency, and λ is wavelength. Rearranging gives λ = v/f. Substituting v = 150 m/s and f = 75 Hz, we find λ = 150/75 = 2.0 m.

A sinusoidal wave is described by the equation

 

𝑦 = 25 sin(100𝜋𝑡 − 3.0𝑥)

 

Where 𝑦 indicates the displacement in 𝑐𝑚, 𝑡 is the time in seconds, and 𝑥 is the distance in meters. Calculate the velocity of the wave.

• 83.3 m s-1

• 75.0 m s-1

• 100.0 m s-1

• 50.0 m s-1

Answer explanation

The wave equation is in the form y = A sin(ωt - kx). The wave velocity v is given by v = ω/k. Here, ω = 100π rad/s and k = 3.0 rad/m. Thus, v = (100π)/(3.0) = 33.33π ≈ 83.3 m/s.

A progressive wave is represented by equation

 

𝑦 = 20 sin(150𝜋𝑡 − 1.8𝑥)

 

Where 𝑦 is the displacement in 𝑐𝑚, 𝑡 is the time in second and 𝑥 is the distance in meter. Calculate the wave velocity.

• 261.8 m s-1

• 211.8 m s-1

• 241.8 m s-1

• 161.8 m s-1

Answer explanation

The wave velocity (v) can be calculated using the formula v = ω/k, where ω = 150π rad/s and k = 1.8 rad/m. Thus, v = (150π)/(1.8) = 261.8 m/s. Therefore, the correct answer is 261.8 m/s.