Customarily in the physics of sound, static-acoustic-related topics are addressed. For instance, the change in the sound level vs discrete change in the distance. In dynamic cases, e.g. the Doppler shit although the relative motion of the components , <i>i.e</i>. the source and the sensor are essential , the movements are limited to uniform motions. In this investigating report, scenarios departed from these limitations are considered. For the former case , time dependent sound level and for the latter case, nonuniform motions are analyzed . Aside from light long-hand mathematical formulations, the majority of the analysis is carried out utilizing a Computer Algebra System (CAS) specifically <i>Mathematica</i>. The analysis and format of the development are crafted flexibly conducive opportunities for furt hering quests for the “what if” scenarios.
Acoustic sound level β measured in dB is given by,
β = 10 log ( I I 0 ) , (1)
whereI0 and I are the reference and sample intensities of the sound, respectively, and both are measured in W/m2. The value of the former being the minimum audible intensity for the human ear is 1 × 10−12 W/m2. In practice, the sample intensity,I, is related to the power of the source, P in watts, via
P = I A , (2)
where A is the surface area through which the sound waves go through. For simplicity assuming the sound waves originate from a point-source spreading evenly throughout the space, i.e. space is homogeneous and isotropic area A is considered a sphere, with a surface area A = 4πR2, with R being its radius.
With these assumptions (1) reads,
β = 10 log ( p 4 π R 2 I 0 ) , (3)
Customary, in practice for a chosen power, P and distance R from the source one measures the sound level β [
With this insight to reach our honed objective, we consider two sound sources and their relative contributing sound levels. This eliminates the explicit need for utilizing I0 and provides a forum to compare the impact of the different source powers.
With these objectives, we craft our report which is composed of three sections. In addition to the Introduction, in Section 2 and its subsections, we develop the needed formulation embodying various scenarios concerning the character of the motions of the sources. Applying a CAS, specifically Mathematica [
Here is the posed problem. Two loudspeakers, with output powers P1 and P2 are a distance d away from a sound sensor. Simultaneously, they put out identical sound notes and begin moving in the same direction with a zero initial speed toward the sensor. In scenario one, source 1 moves at a constant velocity v1, and source 2 at a constant acceleration a2. In scenario two, sources begin accelerating at constant rates a1 and a2, and one is modulated with oscillations. Question: When does the sound-level difference at the sensor reach the maximum?
The second somewhat-related major question concerns the classic Doppler shift, it stems from a classic problem [
Literature search reveals this problem and its solution has never been modified. We altered the posed question by asking: Is it possible to measure the depth of a water well using only a tuning fork? I.e. no timer or no rope! Our solution is insightful.
The forthcoming sections address these issues. Sections 2.1 and 2.2 are concerned with the sound level and 2.3 is the water well problem.
Sources put out power P1 and P2. Applying (3) the difference of the sound level with appropriately changed notations, yields,
Δ β = β 1 − β 2 = 10 [ log ( P 1 P 2 ) + 20 log ( R 12 R 11 ) ] , (4)
According to
{ R 11 = d − v 1 t R 12 = d − 1 2 a 2 t 2 , (5)
Substituting (5) in (4) and assuming Doppler shit does not affect the intensity yields the explicit time-dependent sound-level difference,
Δ β 1 ( t ) = 10 [ log ( P 1 P 2 ) + 20 log ( d − 1 2 a 2 t 2 d − v 1 t ) ] , (6)
One of our objectives is to find the time, t to maximize the sound-level difference. We set the slope of (6) zero and search for its root(s). These are,
Δβ1 [t_] = 10(Log [10, P1/P2] + 20 Log [10,(d − 1/2 a2 t2)/(d − v1t)]);
slopeΔβ1 = D [Δβ1 [t], {t, 1}]//Simplify;
solt1 = Solve [slopeΔβ1 = = 0, t]
{ { t → d a 2 − d 2 a 2 2 − 2 d a 2 v 1 2 a 2 v 1 } , { t → d a 2 + d 2 a 2 2 − 2 d a 2 v 1 2 a 2 v 1 } }
To obtain a realistic meaningful output after trial and error we choose a set of practical parameters storing them in values 1. Units are MKS and symbols correspond one-to-one to the passage in the text.
values 1 = {v1 → 2, a2 → 5, d → 10, P1 → 200, P2 → 30};
solt1/.values1//N
{ { t → 0.4174 } , { t → 9.582 } }
At these time instances, the sources are at the same distances from the sensor yielding the maximum intensity sound-level differences. The distance corresponding to the second time instance is ignored; the distance exceeds d. The acceptable [Δβ(t)]max is,
Δβ1 [t]/.solt1 [[
11.9426 sec
It is insightful to display Δβ(t) side-by-side with the “distance to the sensor” vs t see
plot11 = Plot [Δβ1 [t]/.values1, {t, 0, 1}, AxesLabel → {“t(s)”, “Δβ1(dB)”}, PlotStyle → Black, GridLines → Automatic];
plot12 = Plot [Evaluate [{d − v1 t, d − 1/2 a2 t2}/.values1], {t, 0, 1}, PlotStyle → {Blue, Red}, AxesLabel → {“t(s”, “distance to sensor(m)”}, GridLines → Automatic];
table11 = NumberForm [TableForm [Table[{v1t − 1/2 a2 t2, t}/.values1, {t, 0.3, 0.6, 0.05}], TableHeadings → {None, {“[Δd = v1t − 1/2a2t2](m)”, “t(s)”}}], {3,3}];
The intersecting point of these two curves shown in the middle panel of
{N [Δβ1 [t]/.values1/.t → 0], Δβ1 [t]/.values1/.t → 0.8}
{8.23909, 8.23909}
For identical sources, i.e. for P1 = P2, this leads to Δβ = 0.
One of the objectives of this report is to demonstrate how to design a physics problem. Throughout crafting this report we realized to maximize the sound-level
difference the two sound sources ought to move with different kinematics. In this example, we selected one to move with a constant speed and the other with a constant acceleration. Their character differences are shown on the middle panel of
Intrigued by the learned lesson we extend the design of the physics problem by considering a modified version of the aforementioned case. The details are discussed in Subsection 2.2.
Here both sources are moving at the same constant acceleration, one of them has modified modulated oscillations. Their kinematics is given by,
{ R 21 = 1 2 a 22 t 2 + A sin [ 2 π f t ] R 22 = 1 2 a 22 t 2 , (7)
Inserting (7) in (6) gives,
Δ β 2 [ t _ ] = 10 ( log [ 10 , P 1 P 2 ] + 20 log [ 10 , d − 1 2 a 22 t 2 d − 1 2 a 22 t 2 − A sin [ 2 π f t ] ] ) , (8)
Similar to the previous objective we search for the instance that maximizes the sound-level difference. In fact, because of oscillations, there are multiple such instances. We set the slope of (8) zero and search for its root(s). Theoretically, this sounds, but in practice, it is challenging that even Mathematica is unable to resolve symbolically. Alternatively, for a reasonable set of parameters that are stored in values 2, we solve the problem numerically.
values 2 = {A → 1., f → 1., a22 → 5., P1 → 200., P2 → 30., d → 10.};
The amplitude of the oscillation and its frequency are set at, f = 1 Hz and A = 1, respectively. These quantities are chosen such that they are compatible with the acceleration a22 = 5 m/s2.
Δβ2 [t_] = 10(Log [10, P1/P2] + 20 Log [10, (d − 1/2 a22 t2)/(d − 1/2 a22 t2 − A Sin [2π f t])])/.values2;
plot21 = Show [{Plot [Δβ2 [t], {t, 0, 2}, AxesLabel → {“t(s)”, “Δβ2(dB)”}, PlotStyle → Black, GridLines → Automatic],
Graphics [{Blue, Dashing [0.01], Line [{{0,8.23}, {2.,8.23}}]}]}];
plot22 = Show [{Plot [Evaluate [{d − 1/2 a22 t2 − A Sin [2π f t], d − 1/2 a22 t2}/.values2], {t, 0, 2}, PlotStyle → {Blue, Red}]}, AxesLabel → {“t(s)”, “distance to sensor(m)”}, GridLines → Automatic];
differences are maxima. These are shown by the four intersects of the horizontal dashed line in black shown on the left panel. The ordinate differences between the red and the blue curves are extreme. These are tabulated in the right-most table. Because the frequency of the oscillations is f = 1 Hz this corresponds to the even time-intervals of 0.5 s shown on the left panel.
In the introduction paragraph of Section 2, Procedure, we referred to the classic water well problem. Briefly, this is about measuring the depth of a water well without lowering a rope into the well. Assuming the only available tool is a stop-watch the run-time for a freely dropped stone from the instance a stone is dropped in the well to the time the splash is heard is conducive to the measured depth.
Aiming at the same objective, here we offer an alternate approach that solution doesn’t require a watch; all that is needed is a tuning fork! Our solution employs 1) the principle of the Doppler shift and 2) uses the modified version of the latter. Noting, the standard classic Doppler shift utilizes the change of frequency due to relative motion at a constant speed, a falling tuning fork is falling at a constant acceleration!
The scenario on hand is depicted in
Here is the outline of our solution. Take a tunning fork of frequency f Hz. Drop it in a well of unknown depth h. While it is accelerating and keeps vibrating it emits sound waves that reach back at the edge of the well at frequency f1. The waves also bounce off from the bottom of the well at frequency f2. Both frequencies f1 and f2 are time-dependent and are continuously changing; the Doppler shift. Noting, that the farther the fork drops the faster it moves and at the splash, it moves the fastest corresponding to the highest frequency, (Subscript [f, 2])Max. The reflected frequency at the top of the well produces a sound-beat, B = ( f 2 ) max − f 1 . Knowing the B we formulate a strategy conducive to measuring the depth, h.
The frequency of the sound at the top and the bottom of the well while the fork falls are, respectfully,
f 1 = f ( 1 1 + v 0 v ) , (9)
f 2 = f ( 1 1 − v 0 v ) , (10)
where the instantaneous speed of the fork is, v 0 = 2 g y , the y is the fallen depth, and v is the sound-speed. Utilizing (9) and (10) the number of the beats is,
B = ( f 2 ) max − f 1 , (11)
Rearranging (11) for the depth, h yields,
h = 1 2 g v 2 [ 1 − 2 f B [ ( f B ) 2 + 1 − f B ] ] , (12)
To demonstrate the usefulness of (12) we use the physiological data applied to a normal human ear [
The h in (12) is a two-variable function: h(f, B). For a tunning fork with frequency and beats within the range of 50 Hz ≤ f ≤ 70 Hz and 6 ≤ B ≤ 12 Hz its contour plot is shown on the left panel of
The sound-level changes as a discrete function of distance from the source. This report extends its variation by replacing the discrete distances with distances that are due to the continuous movement of the source. Two such sources with two different kinematics in two different scenarios are considered. The interplay of their respective kinematics utilizing a CAS specifically Mathematica reveals features not reported in the literature. Based on the progress made in this report, similar scenarios considering various kinematics of interest may be analyzed.
This report also revisits the classic “water well” problem. The classic objective of the problem is intact but we offer a different solution providing only a tuning fork instead of a watch.
The interested reader may find a motivational source related to the current study [
The author acknowledges the John T. and Page S. Smith Professorship funds for completing and publishing this work.
The author declares no conflicts of interest regarding the publication of this paper.
Sarafian, H. (2022) Designing Physics Problems with Mathematica Example II. American Journal of Computational Mathematics, 12, 297-305. https://doi.org/10.4236/ajcm.2022.123019