A Bat's Call into Water

Q. A bat emits ultrasonic sound of frequency 1000 kHz in air. Speed of sound in water is 1486 m/s. If the sound meets a water surface, what is the wavelength of the transmitted sound, in metres? expects 0.00149 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 water is 1486 m/s $v_{t} = 1486$ m/s. If the sound meets a water surface $ν_{t} = ν_{c}$ , what is the wavelength of the transmitted sound, in metres?

The playground

Inputs

frequency

ν_{c}

1000 kHz

speed

v_{t}

1486.0 m/s

Outputs

wavelength

λ_{t}

The solution

Solved with TravellingWave

Given

$ν_{c} = 1000$ kHz
$v_{t} = 1486$ m/s
$ν_{t} = ν_{c}$ from “meets a water surface”

Find $λ_{t}$ .

Working

$ν_{c} = 1000$ kHz $= 1 \times 10^{6}$ Hz
meets a water surface $ν_{t} = ν_{c}$
$ν_{t} = 1 \times 10^{6}$
$ν_{t} = 1 \times 10^{6}$ Hz
speed of a wave(transmitted) $v_{t} = ν_{t} \cdot λ_{t}$
$1486 = 1 \times 10^{6} \cdot λ_{t}$
$λ_{t} = 0.001486$ m
$λ_{t} \approx 0.00149$ mto 3 significant figures

Conclusion

Crossing into water the sound keeps the frequency it was made with but runs faster, so each wave stretches longer in the same proportion.

The Ruby

1require_relative "../physics"23question :a_bat_s_call_into_water do4  kind :numerical5  describes "A bat emits ultrasonic sound of frequency %{frequency} in air. Speed of sound in water is %{speed}. If " \6            "the sound meets a water surface, %{wavelength}, in metres?"7  given frequency: "1000 kHz"8  given speed: "1486 m/s"9  asks wavelength: "what is the wavelength of the transmitted sound"10end

1require_relative "../questions/a_bat_s_call_into_water"2require_relative "../science/waves/travelling_wave"34solve :a_bat_s_call_into_water, using: TravellingWave do5  objects :call, :transmitted67  take frequency: call.frequency8  take speed: transmitted.wave_speed910  deduce_from "meets a water surface", transmitted.frequency => call.frequency1112  wavelength = transmitted.wavelength13  answer wavelength:, in: "m", figures: 31415  conclusion "Crossing into water the sound keeps the frequency it was made with but runs faster, so each wave " \16             "stretches longer in the same proportion."17end

1require_relative "../solutions/a_bat_s_call_into_water"23play_with :a_bat_s_call_into_water do4  title "A Bat's Call into 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 transmitted	$k_{t}$	m⁻¹	—
wavelength of transmitted	$λ_{t}$	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 transmitted	$ω_{t}$	rad/s	—
period of transmitted	$T_{t}$	s	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
frequency of call	$ν_{c}$	Hz	0 … ∞
frequency of transmitted	$ν_{t}$	Hz	0 … ∞
wave speed of ground	$v_{g}$	m/s	—
wave speed of call	$v_{c}$	m/s	—
wave speed of transmitted	$v_{t}$	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 transmitted	$Δ φ_{t}$	—	—
separation of transmitted	$Δ x_{t}$	m	—
amplitude of call	$a_{c}$	m	—
amplitude of transmitted	$a_{t}$	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_{t} = \frac{2 \cdot π}{λ_{t}}

wave number and wavelength(transmitted)

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_{t} = \frac{2 \cdot π}{T_{t}}

angular frequency and period(transmitted)

ν_{g} = \frac{1}{T_{g}}

frequency and period(ground)

ν_{c} = \frac{1}{T_{c}}

frequency and period(call)

ν_{t} = \frac{1}{T_{t}}

frequency and period(transmitted)

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_{t} = ν_{t} \cdot λ_{t}

speed of a wave(transmitted)

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_{t} = k_{t} \cdot Δ x_{t}

phase across a separation(transmitted)

ν_{t} = ν_{c}

meets a water surface