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    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–1? The operating frequency of the scanner is 4.2 MHz.

    Speed of sound in the tissue, v = 1.7 km/s = 1.7 × 103m/s

    Operating frequency of the scanner, ν = 4.2 MHz

                                                              = 4.2 × 10    6 Hz

    The wavelength of sound in the tissue is given as,

    λ = v/v

       = 1.7 × 103 / 4.2 × 106 

        = 4.1 × 10    -4 m.
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    Use the formula v = square root of γP over straight rho end root to explain why the speed of sound in air:
    (b) increases with temperature,

    Take the relation, 

    v = √γP/ρ                               ...(i) 

    For one mole of any ideal gas, the equation can be written as: 

    PV = RT 

    P = RT/V                                   ...(ii) 

    Substituting equation (ii) in equation (i), we get: 

    v = √γRT/Vρ  = √γRT/M            ...(iii) 

    where, 

    mass, M = ρV is a constant 

    γ and R are also constants. 

    We conclude from equation (iii) that v ∝ √T 

    Hence, the speed of sound in a gas is directly proportional to the square root of the temperature of the gaseous medium, i.e., the speed of the sound increases with an increase in the temperature of the gaseous medium and vice versa. 
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    Use the formula v = √γP/ρ to explain why the speed of sound in air 

    (c) increases with humidity.

    Let vm and vd be the speed of sound in moist air and dry air respectively.

    Let ρm and ρd be the densities of the moist air and dry air respectively.


     
    However, the presence of water vapour reduces the density of air.

    i.e.,                ρd < ρm 

    ∴                    vm > vd 

    Hence, the speed of sound in moist air is greater than it is in dry air.

    Thus, in gaseous medium, the speed of sound increases with humidity. 
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    Answer
    You have learnt that a travelling wave in one dimension is represented by a function y = f (x, t)where x and t must appear in the combination x – v t or x + v t, i.e.y = f (x ± v t).
    Is the converse true? Examine if the following functions for y can possibly represent a travelling wave:
    (a) (x – vt)2

    (b) log [(x + vt) / x0]

    (c) 1 / (x + vt)
    No, the converse is not true. The basic requirements for a wave function to represent a travelling wave is that for all values of x and t, wave function must have finite value. Out of the given functions for y, no one satisfies this condition. Therefore, none can represent a travelling wave.
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    Answer
    A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of (a) the reflected sound, (b) the transmitted sound? Speed of sound in air is 340 m s–1 and in water 1486 m s–1.
    (a) Frequency of the ultrasonic sound, ν = 1000 kHz = 106 Hz 

    Speed of sound in air, va = 340 m/s 

    The wavelength (λr) of the reflected sound is given by the relation, 

    λr = v/v 

        = 340/106 

        = 3.4 × 10    -4 m.

    (b) Frequency of the ultrasonic sound, ν = 1000 kHz

                                                                 = 10    6 Hz 

    Speed of sound in water, vw = 1486 m/s 

    The wavelength of the transmitted sound is given as, 

    λr = 1486 × 106

            = 1.49 × 10-3 m.
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