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Chapter 4: Problem 40

Many fish have an organ known as a swim bladder, an air-filled cavity whose main purpose is to control the fish's buoyancy an allow it to keep from rising or sinking without having to use its muscles. In some fish, however, the swim bladder (or a small extension of it) is linked to the ear and serves the additional purpose of amplifying sound waves. For a typical fish having such an anatomy, the bladder has a resonant frequency of \(300 \mathrm{~Hz}\), the bladder's \(Q\) is 3 , and the maximum amplification is about a factor of 100 in energy. Over what range of frequencies would the amplification be at least a factor of \(50 ?\)

Short Answer

Expert verified
The amplification is at least 50 in the frequency range from approximately 290 Hz to 310 Hz.

Step by step solution

01

Understand the Given Parameters

We are given that the resonant frequency \( f_r \) of the swim bladder is \( 300 \mathrm{~Hz} \), the quality factor \( Q \) is 3, and the maximum energy amplification factor is 100.
02

Review Quality Factor and Bandwidth

The quality factor \( Q \) is related to the resonant frequency and the bandwidth \( \Delta f \) at which the amplification drops to half the maximum (\( -3 \mathrm{~dB} \)). The relation is \( Q = \frac{f_r}{\Delta f} \).
03

Calculate Bandwidth for Maximum Amplification

Calculate the bandwidth using the formula \( \Delta f = \frac{f_r}{Q} \). Substituting the given values, we have \( \Delta f = \frac{300}{3} = 100 \mathrm{~Hz} \).
04

Determine Amplification at Half Maximum

The bandwidth of \( 100 \mathrm{~Hz} \) represents the frequency range where amplification is \( \frac{1}{\sqrt{2}} \times 100 \approx 70.7 \), which is not our interest here. We want the range where amplification is at least 50.
05

Relate Energy Amplification to Decibels

We know that an amplification factor of 50 is reduced from the peak of 100. In decibels, this means a drop of about \(20 \log_{10} ( \frac{50}{100} ) \approx 6.0206 \mathrm{~dB} \). This requires a smaller bandwidth.
06

Calculate the Required Frequency Range

The frequency range where amplification is at least 50 can be found using the formula for \( Q \) with the new desired drop: \( \Delta f_{new} = \frac{f_r}{Q} \times \frac{(\text{factor for } 6.0206 \mathrm{~dB})}{(\text{factor for } 3 \mathrm{~dB})} \). Calculate this to find the exact limits.

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Key Concepts

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

Buoyancy Control
The swim bladder is a fascinating organ that plays a crucial role in the buoyancy control of fish. Imagine a fish effortlessly floating in water without using its muscles. This magical act is due to the swim bladder. It's an air-filled cavity inside the fish that helps it maintain its position in the water column. Whether it's rising, sinking, or staying still, the swim bladder adjusts the volume of gas inside it to control buoyancy.

Here's how it works effectively:
  • By increasing the gas volume, the fish becomes more buoyant, causing it to ascend.
  • By decreasing the gas volume, the fish becomes less buoyant, allowing it to sink.
  • This ability to control buoyancy with such precision means fish expend minimal energy to move vertically.
This ingenious adaptation allows fish to conserve energy while navigating their aquatic environments.
Sound Amplification
Sound amplification in fish is another remarkable function of the swim bladder. In some species, the swim bladder is connected to the ear, amplifying sound waves much like a speaker enhances music. This specialized capability enhances the fish's hearing, allowing it to detect sound waves underwater that might otherwise be too faint to notice.

This amplification works as follows:
  • The swim bladder resonates with incoming sound waves, magnifying their intensity.
  • These sound waves are then transmitted more effectively to the inner ear.
  • As a result, fish can hear better in the noisy underwater environment, improving survival by detecting predators and prey.
Overall, this added function of the swim bladder demonstrates its versatility and importance in aquatic life.
Resonant Frequency
The resonant frequency is a fundamental concept of sound amplification in the swim bladder. It refers to the specific frequency at which the swim bladder naturally vibrates. In the case of fish, the resonant frequency is around 300 Hz. This means that the swim bladder most effectively resonates with sound waves that match this frequency.

Think of it like this:
  • When sound waves hit the swim bladder at this particular frequency, they cause the bladder to vibrate strongly.
  • The vibration enhances the sound, making it louder and more noticeable to the fish.
  • This frequency is crucial for maximizing the efficiency of sound amplification.
Understanding resonant frequency helps explain how the swim bladder amplifies sound, playing a pivotal role in the fish's acoustic perception.
Quality Factor
The quality factor, often denoted as "Q," refers to how effectively a system resonates at its natural frequency. It is a crucial parameter when discussing sound amplification and resonance in the swim bladder.

Here's why it's important:
  • The quality factor is calculated as the ratio of the resonant frequency to the bandwidth (\( Q = \frac{f_r}{\Delta f} \)).
  • A higher Q indicates a sharper, more selective resonance peak.
  • A Q of 3, for example, means the swim bladder has a moderate ability to resonate, allowing it to amplify only those frequencies close to the resonant frequency.
By understanding Q, we can predict the range of frequencies where amplification is maximized, which is vital for determining how well fish can detect sounds in their environment.

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Most popular questions from this chapter

(a) Let \(T\) be the maximum tension that an elevator's cable can withstand without breaking, i.e., the maximum force it can exert. If the motor is programmed to give the car an acceleration \(a(a>0\) is upward), what is the maximum mass that the car can have, including passengers, if the cable is not to break?(answer check available at lightandmatter.com) (b) Interpret the equation you derived in the special cases of \(a=0\) and of a downward acceleration of magnitude \(g\).

A helicopter of mass \(m\) is taking off vertically. The only forces acting on it are the earth's gravitational force and the force, \(F_{a i r}\), of the air pushing up on the propeller blades. (a) If the helicopter lifts off at \(t=0\), what is its vertical speed at time \(t\) ? (b) Check that the units of your answer to part a make sense. (c) Discuss how your answer to part a depends on all three variables, and show that it makes sense. That is, for each variable, discuss what would happen to the result if you changed it while keeping the other two variables constant. Would a bigger value give a smaller result, or a bigger result? Once you've figured out this mathematical relationship, show that it makes sense physically. (d) Plug numbers into your equation from part a, using \(m=2300 \mathrm{~kg}, F_{a i r}=27000 \mathrm{~N}\), and \(t=4.0 \mathrm{~s}\). (answer check available at lightandmatter.com)

A plane is flown in a loop-the-loop of radius \(1.00 \mathrm{~km}\). The plane starts out flying upside-down, straight and level, then begins curving up along the circular loop, and is right-side up when it reaches the top. (The plane may slow down somewhat on the way up.) How fast must the plane be going at the top if the pilot is to experience no force from the seat or the seatbelt while at the top of the loop? (answer check available at lightandmatter.com)

A person on a bicycle is to coast down a ramp of height \(h\) and then pass through a circular loop of radius \(r\). What is the smallest value of \(h\) for which the cyclist will complete the loop without falling? (Ignore the kinetic energy of the spinning wheels.)(answer check available at lightandmatter.com)

Annie Oakley, riding north on horseback at \(30 \mathrm{mi} / \mathrm{hr}\), shoots her rifle, aiming horizontally and to the northeast. The muzzle speed of the rifle is \(140 \mathrm{mi} / \mathrm{hr}\). When the bullet hits a defenseless fuzzy animal, what is its speed of impact? Neglect air resistance, and ignore the vertical motion of the bullet.(solution in the pdf version of the book)
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