A Bat's Call Reflected off Water

solved TravellingWave

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?

Inputs

frequency

ν_{c}

1000 kHz

speed

v_{c}

340.0 m/s

Outputs

wavelength

λ_{e}

How it's solved

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 web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

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

solution.rb

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

playground.rb

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 it drew on

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)