A Bat's Call Reflected off Water

Q. A bat emits ultrasonic sound of frequency 1000 kHz in air. Speed of sound in air is 340 m/s. If the sound meets a water surface, what is the wavelength of the reflected sound, in metres? expects 3.4 × 10⁻⁴ m

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Q. A bat emits ultrasonic sound of frequency 1000 kHz $ν_{c} = 1000$ kHz in air. Speed of sound in air is 340 m/s $v_{c} = 340$ m/s. If the sound meets a water surface, what is the wavelength of the reflected sound $ν_{e} = ν_{c}$ , $v_{e} = v_{c}$ , in metres?

The playground

Inputs

frequency

ν_{c}

1000 kHz

speed

v_{c}

340.0 m/s

Outputs

wavelength

λ_{e}

The solution

Solved with TravellingWave

Given

$ν_{c} = 1000$ kHz
$v_{c} = 340$ m/s
$ν_{e} = ν_{c}$ from “the reflected sound”
$v_{e} = v_{c}$ from “the reflected sound”

Find $λ_{e}$ .

Working

$ν_{c} = 1000$ kHz $= 1 \times 10^{6}$ Hz
the reflected sound (wave speed of echo) $v_{e} = v_{c}$
$v_{e} = 340$
$v_{e} = 340$ m/s
the reflected sound (frequency of echo) $ν_{e} = ν_{c}$
$ν_{e} = 1 \times 10^{6}$
$ν_{e} = 1 \times 10^{6}$ Hz
speed of a wave(echo) $v_{e} = ν_{e} \cdot λ_{e}$
$340 = 1 \times 10^{6} \cdot λ_{e}$
$λ_{e} = 3.4 \times 10^{−4}$ m
$λ_{e} \approx 3.4 \times 10^{−4}$ mto 2 significant figures

Conclusion

The reflected sound goes back into the same air at the same frequency, so its wavelength is the speed in air over that frequency.

The Ruby

1require_relative "../physics"23question :a_bat_s_call_reflected_off_water do4  kind :numerical5  describes "A bat emits ultrasonic sound of frequency %{frequency} in air. Speed of sound in air is %{speed}. If " \6            "the sound meets a water surface, %{wavelength}, in metres?"7  given frequency: "1000 kHz"8  given speed: "340 m/s"9  asks wavelength: "what is the wavelength of the reflected sound"10end

1require_relative "../questions/a_bat_s_call_reflected_off_water"2require_relative "../science/waves/travelling_wave"34solve :a_bat_s_call_reflected_off_water, using: TravellingWave do5  objects :call, :echo67  take frequency: call.frequency8  take speed: call.wave_speed910  deduce_from "the reflected sound", echo.frequency => call.frequency, echo.wave_speed => call.wave_speed1112  wavelength = echo.wavelength13  answer wavelength:, in: "m", figures: 21415  conclusion "The reflected sound goes back into the same air at the same frequency, so its wavelength is the " \16             "speed in air over that frequency."17end

1require_relative "../solutions/a_bat_s_call_reflected_off_water"23play_with :a_bat_s_call_reflected_off_water do4  title "A Bat's Call Reflected off Water"5end

The laws

quantity	written	units	within	from
pi	$π$	—	0 … ∞	TravellingWave
amplitude	$a$	m	—	TravellingWave
wave number	$k$	m⁻¹	—	TravellingWave
angular frequency	$ω$	rad/s	—	TravellingWave
wavelength	$λ$	m	0 … ∞	TravellingWave
period	$T$	s	—	TravellingWave
frequency	$ν$	Hz	0 … ∞	TravellingWave
wave speed	$v$	m/s	—	TravellingWave
separation	$Δ x$	m	—	TravellingWave
phase difference	$Δ φ$	—	—	TravellingWave
wave number of ground	$k_{g}$	m⁻¹	—
wavelength of ground	$λ_{g}$	m	0 … ∞
wave number of call	$k_{c}$	m⁻¹	—
wavelength of call	$λ_{c}$	m	0 … ∞
wave number of echo	$k_{e}$	m⁻¹	—
wavelength of echo	$λ_{e}$	m	0 … ∞
angular frequency of ground	$ω_{g}$	rad/s	—
period of ground	$T_{g}$	s	—
angular frequency of call	$ω_{c}$	rad/s	—
period of call	$T_{c}$	s	—
angular frequency of echo	$ω_{e}$	rad/s	—
period of echo	$T_{e}$	s	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
frequency of call	$ν_{c}$	Hz	0 … ∞
frequency of echo	$ν_{e}$	Hz	0 … ∞
wave speed of ground	$v_{g}$	m/s	—
wave speed of call	$v_{c}$	m/s	—
wave speed of echo	$v_{e}$	m/s	—
phase difference of ground	$Δ φ_{g}$	—	—
separation of ground	$Δ x_{g}$	m	—
phase difference of call	$Δ φ_{c}$	—	—
separation of call	$Δ x_{c}$	m	—
phase difference of echo	$Δ φ_{e}$	—	—
separation of echo	$Δ x_{e}$	m	—
amplitude of call	$a_{c}$	m	—
amplitude of echo	$a_{e}$	m	—

k_{w} = \frac{2 \cdot π}{λ_{w}}

wave number and wavelength

ω_{w} = \frac{2 \cdot π}{T_{w}}

angular frequency and period

ν_{w} = \frac{1}{T_{w}}

frequency and period

v_{w} = ν_{w} \cdot λ_{w}

speed of a wave

Δ φ_{w} = k_{w} \cdot Δ x_{w}

phase across a separation

k_{g} = \frac{2 \cdot π}{λ_{g}}

wave number and wavelength(ground)

k_{c} = \frac{2 \cdot π}{λ_{c}}

wave number and wavelength(call)

k_{e} = \frac{2 \cdot π}{λ_{e}}

wave number and wavelength(echo)

ω_{g} = \frac{2 \cdot π}{T_{g}}

angular frequency and period(ground)

ω_{c} = \frac{2 \cdot π}{T_{c}}

angular frequency and period(call)

ω_{e} = \frac{2 \cdot π}{T_{e}}

angular frequency and period(echo)

ν_{g} = \frac{1}{T_{g}}

frequency and period(ground)

ν_{c} = \frac{1}{T_{c}}

frequency and period(call)

ν_{e} = \frac{1}{T_{e}}

frequency and period(echo)

v_{g} = ν_{g} \cdot λ_{g}

speed of a wave(ground)

v_{c} = ν_{c} \cdot λ_{c}

speed of a wave(call)

v_{e} = ν_{e} \cdot λ_{e}

speed of a wave(echo)

Δ φ_{g} = k_{g} \cdot Δ x_{g}

phase across a separation(ground)

Δ φ_{c} = k_{c} \cdot Δ x_{c}

phase across a separation(call)

Δ φ_{e} = k_{e} \cdot Δ x_{e}

phase across a separation(echo)

ν_{e} = ν_{c}

the reflected sound (frequency of echo)

v_{e} = v_{c}

the reflected sound (wave speed of echo)