A Harmonic Wave's Frequency

Q. A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36 t + 0.018 x + π/4) where x and y are in cm and t in s. What is its frequency, in hertz? expects 5.7 Hz

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Q. A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36 t $ω = 36$ rad/s + 0.018 x + π/4) where x and y are in cm and t in s. What is its frequency, in hertz?

The playground

Outputs

frequency

ν

The solution

Solved with TravellingWave

Given

$ω = 36$ rad/s from “36 t”

Find $ν$ .

Working

angular frequency and period(wave) $ω = \frac{2 \cdot π}{T}$
$36 = \frac{2 \cdot 3.14159265}{T}$
$T = 0.174532925$ s
frequency and period(wave) $ν = \frac{1}{T}$
$ν = \frac{1}{0.174532925}$
$ν = 5.73$ Hz
$ν \approx 5.7$ Hzto 2 significant figures

Conclusion

The number multiplying t is the angular frequency, radians of phase each second; a full turn of phase is one oscillation.

The Ruby

1require_relative "../physics"23question :a_harmonic_wave_s_frequency do4  kind :numerical5  describes "A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36 t + 0.018 x + π/4) where x " \6            "and y are in cm and t in s. %{frequency}, in hertz?"7  asks frequency: "What is its frequency"8end

1require_relative "../questions/a_harmonic_wave_s_frequency"2require_relative "../science/waves/travelling_wave"34solve :a_harmonic_wave_s_frequency, using: TravellingWave do5  objects :wave67  deduce_from "36 t", wave.angular_frequency => "36 rad/s"89  frequency = wave.frequency10  answer frequency:, in: "Hz", figures: 21112  conclusion "The number multiplying t is the angular frequency, radians of phase each second; a full turn of " \13             "phase is one oscillation."14end

1require_relative "../solutions/a_harmonic_wave_s_frequency"23play_with :a_harmonic_wave_s_frequency do4  title "A Harmonic Wave's Frequency"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 wave	$k$	m⁻¹	—
wavelength of wave	$λ$	m	0 … ∞
angular frequency of ground	$ω_{g}$	rad/s	—
period of ground	$T_{g}$	s	—
angular frequency of wave	$ω$	rad/s	—
period of wave	$T$	s	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
frequency of wave	$ν$	Hz	0 … ∞
wave speed of ground	$v_{g}$	m/s	—
wave speed of wave	$v$	m/s	—
phase difference of ground	$Δ φ_{g}$	—	—
separation of ground	$Δ x_{g}$	m	—
phase difference of wave	$Δ φ$	—	—
separation of wave	$Δ x$	m	—
amplitude of wave	$a$	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 = \frac{2 \cdot π}{λ}

wave number and wavelength(wave)

ω_{g} = \frac{2 \cdot π}{T_{g}}

angular frequency and period(ground)

ω = \frac{2 \cdot π}{T}

angular frequency and period(wave)

ν_{g} = \frac{1}{T_{g}}

frequency and period(ground)

ν = \frac{1}{T}

frequency and period(wave)

v_{g} = ν_{g} \cdot λ_{g}

speed of a wave(ground)

v = ν \cdot λ

speed of a wave(wave)

Δ φ_{g} = k_{g} \cdot Δ x_{g}

phase across a separation(ground)

Δ φ = k \cdot Δ x

phase across a separation(wave)