A Tsunami Slows as the Water Shallows, and Its Height Grows as the Inverse Fourth Root of the Depth ── A tsunami travelling at 198.057062 m/s (713 km/h) over 4000 m of water slows to 9.902853 m/s in 10 m ── solving the linear shallow-water equations numerically agrees on the slope to within 1.6 per cent ── [Paper 616]

Abstract

How the speed of an ocean wave is set changes according to whether its wavelength is much longer or much shorter than the depth. The speed of a tsunami and the growth of its height in shallow water are obtained by two routes, formula and numerical computation. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──the shallow-water wave speed sqrt(gh), the law for the height when the depth changes (Green, 1838), and the dispersion relation of waves (in Lamb's textbook and elsewhere) are all known. Breaking and run-up are not treated──only linear waves whose height is small compared with the depth are considered; the height reached on land is not stated. Actual tsunami events are not treated──arrival times and recorded heights are not discussed; depths and periods are only set and computed. The cause of the discrepancy on the shelf has not been checked──part of the wave may be reflected at the kink at the end of the slope, but this is kept as a hypothesis. Relation to earlier papers: Paper 391 showed that the significant wave height is neither the mean nor the maximum, and left open in its scope note that "breaking and shoaling are not the subject"──this paper treats, within shoaling, the growth of a tsunami's height. What is added is setting tsunamis and wind waves side by side through numerical solution of the dispersion relation, matching Green's law with a numerical solution of the linear shallow-water equations, showing in numbers where it fits and where it departs, and placing the separator on whether the wavelength is longer than the depth. First, a tsunami's speed is fixed by the depth alone──a wave much longer than the depth travels at sqrt(gh): 198.057062 m/s (713 km/h) at 4000 m, 31.315571 m/s at 100 m and 9.902853 m/s at 10 m. Crossing 10000 km of 4000-m ocean takes 14.025139 hours (Section 2). Second, solving the dispersion relation gives the same──solving the general relation for wave speed numerically, a tsunami with a 10-minute period at 4000 m has a wavelength of 117.947696 km and a speed 0.992540 times sqrt(gh). A wind wave with a 10-second period has a wavelength of 156 m, and its speed of 15.607768 m/s is fixed by the period alone, not the depth (Section 2). Third, and this is the core. As the water shallows, the height grows as the inverse 1/4 power of the depth──the flow of energy carried by the wave (the square of the height times the speed) is kept, so the height grows as much as the speed falls (Green's law): 1.414214 times from 4000 m to 1000 m, 2.514867 times to 100 m and 4.472136 times to 10 m, while the wavelength shrinks in proportion to sqrt(h) (Section 3). Fourth, numerical computation agrees with Green's law on the slope to within 1.6 per cent──for a sea shoaling from 4000 m to 40 m over 600 km, solving the linear shallow-water equations on a 6000-point grid gave height ratios of 1.422233 at 1000 m and 2.555027 at 100 m, relative differences from Green's law of 0.005671 to 0.015969. On the shelf beyond the slope (40 m) it gave 3.101584, 0.019193 below (the cause has not been checked) (Section 4). Fifth, the separator is whether the wavelength is longer than the depth──if longer, the speed is fixed by the depth alone, and the wave slows and grows taller as the water shallows. If shorter, the speed is fixed by the period alone and the wave does not feel the depth of the deep ocean (Section 5). A tsunami is a wave far longer than the depth. Its speed is fixed by the depth alone and falls as the water shallows──198.057062 m/s at 4000 m and 9.902853 m/s at 10 m. Solving the dispersion relation, a 10-minute tsunami travels at 0.992540 times sqrt(gh), while a 10-second wind wave's speed is fixed by its period regardless of depth. As the water shallows, the height grows as the inverse 1/4 power of the depth──because the flow of energy carried is kept, the height grows 2.514867 times from 4000 m to 100 m. A numerical solution of the linear shallow-water equations agreed with this on the slope to within 1.6 per cent and fell 0.019193 below it on the shelf beyond. The separator is whether the wavelength is longer than the depth──if longer, the wave feels the depth, slows and grows taller; if shorter, it does not feel the floor of the deep sea. Placed among the earlier papers──it takes the seat of shoaling left open in the scope note of Paper 391. To be honest──breaking, run-up and actual events are not treated, and the cause of the shortfall on the shelf has not been checked. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: tsunami, shallow-water wave, Green's law, wave speed, water depth. ----- 海の波の速さは、波長が水深よりずっと長いか短いかで、決まり方が変わる。津波の速さと、浅い所へ来たときの高さの伸びを、式と数値計算の二つの道筋で出す。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──浅い水の波の速さ sqrt(gh)、水深が変わるときの高さの法則(グリーン、1838)、波の分散の式(ラムの教科書ほか)は、いずれも既知である。砕波・遡上を扱わない──高さが水深に比べて小さい線形の波だけを見る。岸に上がる高さは述べない。実際の津波の事例を扱わない──到達時間や高さの記録は論じず、水深と周期を置いて計算するだけである。棚での食い違いの原因を確かめていない──斜面の終わりの折れ目で波の一部が跳ね返ることが考えられるが、仮説にとどめる。既刊との関係:論文391 は有義波高が平均でも最大でもないことを示し、射程注記で「砕波や浅水変形は主題ではない」と空けていた──本稿はその浅水変形のうち、津波の高さの伸びを扱う。加えたのは、分散の式の数値解で津波と風の波の違いを並べたこと、グリーンの法則を線形の浅水方程式の数値計算と突き合わせ、合う所と外れる所を数で示したこと、分離子を「波長が水深より長いか」に置いたことである。 第一に、津波の速さは水深だけで決まる──波長が水深よりずっと長い波の速さは sqrt(gh) で、水深 4000 m で秒速 198.057062 m(時速 713 km)、100 m で 31.315571 m、10 m で 9.902853 m である。4000 m の海を 10000 km 渡るのに 14.025139 時間かかる(第2節)。 第二に、分散の式を解いても同じである──波の速さの一般の式を数値で解くと、水深 4000 m で周期 10 分の津波は波長 117.947696 km、速さは sqrt(gh) の 0.992540 倍だった。周期 10 秒の風の波は波長 156 m で、速さ 15.607768 m/s は水深によらず周期だけで決まる(第2節)。 第三に、これが本稿の芯である。浅くなると、高さは水深の 1/4 乗に反比例して伸びる──波が運ぶエネルギーの流れ(高さの二乗と速さの積)が保たれるので、速さが落ちた分だけ高さが伸びる(グリーンの法則)。4000 m から 1000 m で 1.414214 倍、100 m で 2.514867 倍、10 m で 4.472136 倍で、波長は sqrt(h) に比例して縮む(第3節)。 第四に、数値計算は、斜面の上でグリーンの法則に 1.6 パーセント以内で合う──水深 4000 m から 40 m へ 600 km かけて浅くなる海で、線形の浅水方程式を 6000 点の格子で解くと、高さの倍率は 1000 m で 1.422233、100 m で 2.555027 で、グリーンの法則との相対差は 0.005671 から 0.015969 だった。斜面を上り切った棚(40 m)では 3.101584 で、0.019193 下回った(原因は確かめていない)(第4節)。 第五に、分離子は、波長が水深より長いかどうかである──長ければ速さは水深だけで決まり、浅くなると遅くなって背が伸びる。短ければ速さは周期だけで決まり、深い海の水深を感じない(第5節)。 津波は、波長が水深よりずっと長い波である。その速さは水深だけで決まり、浅くなるほど遅くなる──水深 4000 m で秒速 198.057062 m、10 m で 9.902853 m である。分散の式を解いても、周期 10 分の津波の速さは sqrt(gh) の 0.992540 倍で、周期 10 秒の風の波は水深によらず周期だけで決まった。浅くなると、高さは水深の 1/4 乗に反比例して伸びる──運ばれるエネルギーの流れが保たれるからで、4000 m から 100 m で 2.514867 倍になる。線形の浅水方程式の数値計算は、斜面の上でこれに 1.6 パーセント以内で合い、斜面を上り切った棚では 0.019193 下回った。分離子は、波長が水深より長いかどうかである──長ければ水深を感じて遅くなり背が伸び、短ければ深い海の底を感じない。既刊との位置──論文391 が射程注記で空けた浅水変形の席に座る。正直に言えば──砕波も遡上も実際の事例も扱わず、棚で下回った原因も確かめていない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:津波、浅水波、グリーンの法則、波の速さ、水深。