A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is $1.7\ \text{km s}^{-1}$? The operating frequency of the scanner is $4.2\ \text{MHz}$.

> [!TLDR] > The wavelength is found using the wave relation $v = f\lambda$. Given $v = 1.7\ \text{km/s}$ and $f = 4.2\ \text{MHz}$, the wavelength in the tissue is $\boxed{4.05 \times 10^{-4}\ \text{m}}$ (or $0.405\ \text{mm}$). The core idea here is the universal wave equation: for any wave, speed equals frequency times wavelength. In medical ultrasound, the scanner sends high-frequency sound waves into the body. The wavelength determines how finely the scanner can resolve details — shorter wavelengths give better resolution but penetrate less deeply. The problem gives you the speed in the tissue and the operating frequency, so finding the wavelength is a direct application of $v = f\lambda$. Let’s work through it step by step. 1. **Write down the wave relation.** For any periodic wave, $$v = f \lambda$$ where $v$ is the wave speed, $f$ is the frequency, and $\lambda$ is the wavelength. 2. **Convert all quantities to consistent SI units.** The speed is given as $1.7\ \text{km s}^{-1}$. Since $1\ \text{km} = 1000\ \text{m}$, $$v = 1.7 \times 1000 = 1700\ \text{m s}^{-1}$$ The frequency is $4.2\ \text{MHz}$. Recall $1\ \text{MHz} = 10^6\ \text{Hz}$, so $$f = 4.2 \times 10^6\ \text{Hz}$$ > [!WARNING] > A common mistake is forgetting to convert MHz to Hz or km/s to m/s. If you plug in $4.2$ for frequency and $1.7$ for speed, you’ll get a wavelength of about $0.4\ \text{m}$ — which is absurd for ultrasound in tissue (that’s the size of a football!). Always check units before calculating. 3. **Rearrange the formula to solve for wavelength.** $$\lambda = \frac{v}{f}$$ 4. **Substitute the values.** $$\lambda = \frac{1700}{4.2 \times 10^6}$$ …

A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the ti

Exercises · 14.7

1.7\ \text{km s}^{-1}

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Concept understanding — Wave Speed on String

Wave Speed on a String – From Intuition to Formula

Imagine you and a friend hold a long, taut rope between you. If you give your end a quick flick upward, a bump travels along the rope toward your friend. That bump is a wave, and the speed at which it moves is the wave speed.

Now ask yourself: what determines how fast that bump travels? Two things stand out from everyday experience:

Tension – If you pull the rope tighter, the bump zips along faster. A loose rope makes the wave crawl.
Mass – If the rope is heavy (like a thick clothesline), the wave moves slower than on a light, thin string under the same tension.

So wave speed increases with tension and decreases with the "heaviness" of the string. That's the core intuition.

The Precise Statement

For a wave traveling along a stretched string, the wave speed $v$ is given by:

v = \sqrt{\frac{T}{\mu}}

where:

$T$ is the tension in the string (in newtons, N)
$\mu$ is the linear mass density – the mass per unit length of the string (in kg/m)

v = \sqrt{\frac{T}{\mu}}

This formula is exact for an ideal string (perfectly flexible, no stiffness, no damping). It comes from solving the wave equation for a string, but you can understand it physically.

Why the Square Root? A Quick Physical Argument

Think of a small segment of the string. The tension provides the restoring force that tries to straighten the string when it's bent. A higher tension means a stronger restoring force, so the wave accelerates faster – hence higher speed.

The mass per unit length $\mu$ is the inertia of the string. A heavier string resists acceleration more, so the wave slows down.

The square root appears because the relationship between force, mass, and acceleration isn't linear when you derive it properly. But the key takeaway is:

Important

Wave speed on a string depends only on the string's tension and its linear density – not on the frequency or amplitude of the wave.

This is a surprising and important result. Whether you send a slow, gentle ripple or a fast, sharp pulse, both travel at the same speed on the same string.

A Simple Example

A steel guitar string has $\mu = 0.002\ \text{kg/m}$ and is under tension $T = 100\ \text{N}$ . What is the wave speed?

v = \sqrt{\frac{100}{0.002}} = \sqrt{50000} \approx 224\ \text{m/s}

That's about half the speed of sound in air – fast enough that the wave reaches the other end almost instantly.

Common Mistakes to Avoid

Watch out

Do not confuse wave speed with the speed of the string's particles. The string itself moves up and down (transverse motion), but the wave travels horizontally. These are different speeds.
Wave speed does NOT depend on frequency. Changing how fast you flick your hand changes the frequency, but the wave still travels at $v = \sqrt{T/\mu}$ .
Tension is not the same as force applied at the end. If the string is under tension $T$ everywhere (ideal case), that's the value you use – not the force you apply to create the wave.

Where This Formula Comes From (A Glimpse) …

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The wavelength is found using the wave relation $v = f\lambda$ . Given $v = 1.7\ \text{km/s}$ and $f = 4.2\ \text{MHz}$ , the wavelength in the tissue is $\boxed{4.05 \times 10^{-4}\ \text{m}}$ (or $0.405\ \text{mm}$ ).

The core idea here is the universal wave equation: for any wave, speed equals frequency times wavelength. In medical ultrasound, the scanner sends high-frequency sound waves into the body. The wavelength determines how finely the scanner can resolve details — shorter wavelengths give better resolution but penetrate less deeply. The problem gives you the speed in the tissue and the operating frequency, so finding the wavelength is a direct application of $v = f\lambda$ .

Let’s work through it step by step.

Write down the wave relation. For any periodic wave,

v = f \lambda

where $v$ is the wave speed, $f$ is the frequency, and $\lambda$ is the wavelength.

Convert all quantities to consistent SI units. The speed is given as $1.7\ \text{km s}^{-1}$ . Since $1\ \text{km} = 1000\ \text{m}$ ,

v = 1.7 \times 1000 = 1700\ \text{m s}^{-1}

The frequency is $4.2\ \text{MHz}$ . Recall $1\ \text{MHz} = 10^6\ \text{Hz}$ , so

f = 4.2 \times 10^6\ \text{Hz}

Watch out

A common mistake is forgetting to convert MHz to Hz or km/s to m/s. If you plug in $4.2$ for frequency and $1.7$ for speed, you’ll get a wavelength of about $0.4\ \text{m}$ — which is absurd for ultrasound in tissue (that’s the size of a football!). Always check units before calculating.

Rearrange the formula to solve for wavelength.

\lambda = \frac{v}{f}

Substitute the values. $\lambda = \frac{1700}{4.2 \times 10^6}$ …

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Q.A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km s−11.7\ \text{km s}^{-1}1.7 km s−1? The operating frequency of the scanner is 4.2 MHz4.2\ \text{MHz}4.2 MHz.

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Q.A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is $1.7\ \text{km s}^{-1}$ ? The operating frequency of the scanner is $4.2\ \text{MHz}$ .

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