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Chapter 16: Problem 32

To navigate, a porpoise emits a sound wave that has a wavelength of 1.5 cm. The speed at which the wave travels in seawater is 1522 m/s. Find the period of the wave.

Short Answer

Expert verified
The period of the wave is approximately \(9.86 \times 10^{-6}\) seconds.

Step by step solution

01

Convert Wavelength to Meters

First, change the wavelength from centimeters to meters, because the speed of the wave is given in meters per second. 1 cm = 0.01 m, thus, \(1.5 \text{ cm} = 0.015 \text{ m}\).
02

Recall the Wave Speed Formula

The speed of a wave is determined by the product of its wavelength \(\lambda\) and frequency \(f\), given by the formula: \[ v = \lambda \times f \] Where \(v\) is the speed of the wave, \(\lambda\) is the wavelength, and \(f\) is the frequency.
03

Solve for Frequency

Rearrange the wave speed formula to find the frequency: \[ f = \frac{v}{\lambda} \] Substitute the known values into the formula: \( v = 1522 \, \text{m/s} \) and \( \lambda = 0.015 \, \text{m} \)\[ f = \frac{1522}{0.015} \approx 101466.67 \text{ Hz}\]
04

Calculate the Period

The period \(T\) of a wave is the reciprocal of the frequency, given by: \[ T = \frac{1}{f} \] Substitute the frequency found in Step 3: \[ T \approx \frac{1}{101466.67} \approx 9.86 \times 10^{-6} \text{ seconds} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wavelength Conversion
When working with problems involving waves, it's essential to ensure all units align correctly. In the given exercise, the wavelength of the sound wave is initially provided in centimeters. To work with the wave speed, which is in meters per second, it's necessary to convert the wavelength to meters as well. Here's a quick way to convert:
  • Remember that 1 centimeter equals 0.01 meters.
  • To convert 1.5 cm to meters, simply multiply by the conversion factor: \(1.5 \text{ cm} \times 0.01 = 0.015 \text{ m}\).
This step is crucial for accurate calculations, as mixing units can lead to errors. Keeping consistent units helps maintain precision and clarity throughout the problem-solving process.
Wave Speed Formula
Sound waves, much like other types of waves, obey the wave speed formula. This formula connects the wave's speed, wavelength, and frequency. The relationship is expressed as:\[ v = \lambda \times f \]In this formula:- \(v\) is the wave's speed,- \(\lambda\) is the wavelength,- \(f\) is the frequency.Understanding this formula is vital for solving most wave-related problems. Each of these variables plays a role in defining a wave's behavior. Given any two of these components, you can determine the third. For example, when provided with speed and wavelength, calculating frequency becomes straightforward. This formula is fundamental in physics and helps describe how waves, whether sound or light, travel through different mediums.
Frequency Calculation
Calculating the frequency of a wave is a common task in physics, especially when exploring aspects like sound. When given the speed and wavelength, frequency can be calculated using:\[ f = \frac{v}{\lambda} \]This particular exercise provides:- A speed of \(1522 \text{ m/s}\)- A wavelength of \(0.015 \text{ m}\)By substituting these values into the formula, the frequency is determined as follows: \[ f = \frac{1522}{0.015} \approx 101466.67 \text{ Hz} \]Frequency has the unit Hertz (Hz), which signifies cycles per second. A high frequency indicates more cycles per second, whereas a low frequency means fewer cycles.This step is notable because frequency affects many characteristics of a wave like sound pitch or color in light.
Seawater Sound Wave Properties
Sound waves behave differently in seawater compared to air due to variations in density and other factors. In general, sound travels faster in seawater because it is a denser medium. Such differences dramatically influence aspects like speed and wavelength.In this exercise, the speed of sound in seawater is given as \(1522 \text{ m/s}\). This is typical for sound waves in such a medium, highlighting the efficiency of sound travel under water.Understanding seawater's properties also helps in numerous practical applications:
  • Navigation for marine animals like porpoises relies heavily on sound waves.
  • Submarine and underwater communication systems also leverage these properties of seawater to transmit signals across long distances.
Grasping the nuances of sound wave travel through seawater provides insights into both natural and technological phenomena.

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Most popular questions from this chapter

Two wires are parallel, and one is directly above the other. Each has a length of 50.0 m and a mass per unit length of 0.020 kg/m. However, the tension in wire \(A\) is \(6.00 \times 10^{2} \mathrm{N},\) and the tension in wire \(\mathrm{B}\) is \(3.00 \times 10^{2} \mathrm{N}\) . Transverse wave pulses are generated simultaneously, one at the left end of wire A and one at the right end of wire B. The pulses travel toward each other. How much time does it take until the pulses pass each other?

Tsunamis are fast-moving waves often generated by underwater earth- quakes. In the deep ocean their amplitude is barely noticeable, but upon reaching shore, they can rise up to the astonishing height of a six-story building. One tsunami, generated off the Aleutian islands in Alaska, had a wavelength of 750 km and traveled a distance of 3700 km in 5.3 h. (a) What was the speed (in m/s) of the wave? For reference, the speed of a 747 jetliner is about 250 m/s. Find the wave鈥檚 (b) frequency and (c) period.

Hearing damage may occur when a person is exposed to a sound intensity level of 90.0 dB (relative to the threshold of hearing) for a period of 9.0 hours. One particular eardrum has an area of \(2.0 \times 10^{-4} \mathrm{m}^{2} .\) How much sound energy is incident on this eardrum during this time?

Using an intensity of \(1 \times 10^{-12} \mathrm{W} / \mathrm{m}^{2}\) as a reference, the threshold of hearing for an average young person is 0 \(\mathrm{dB}\) . Person 1 and person \(2,\) who are not average, have thresholds of hearing that are \(\beta_{1}=-8.00 \mathrm{dB}\) and \(\beta_{2}=+12.0 \mathrm{dB}\) . What is the ratio \(I_{1} / I_{2}\) of the sound intensity \(I_{1}\) when person 1 hears the sound at his own threshold of hearing compared to the sound intensity \(I_{2}\) when person 2 hears the sound at his own threshold of hearing?

The speed of a sound in a container of hydrogen at 201 K is 1220 m/s. What would be the speed of sound if the temperature were raised to 405 K? Assume that hydrogen behaves like an ideal gas.
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